answer：To determine the coefficient of the ( x^3 ) term in the expansion of ( (1 + 2x)^5 ), I can use the binomial theorem. The binomial theorem states that ( (a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n - k} b^k ).In this case, ( a = 1 ), ( b = 2x ), and ( n = 5 ). I'm interested in the term where the power of ( x ) is 3, which corresponds to ( k = 3 ).First, I calculate the binomial coefficient ( binom{5}{3} ), which is 10.Next, I raise ( a ) to the power of ( 5 - 3 = 2 ), resulting in ( 1^2 = 1 ).Then, I raise ( b ) to the power of 3, which gives ( (2x)^3 = 8x^3 ).Finally, I multiply these results together: ( 10 times 1 times 8x^3 = 80x^3 ).Therefore, the coefficient of the ( x^3 ) term is 80.

answer：Okay, so I have this problem where I need to find the value of (2a - b) · a, given that the angle between vectors a and b is 120 degrees, and their magnitudes are |a| = 2 and |b| = 5. Hmm, let me think about how to approach this.First, I remember that the dot product of two vectors can be calculated using the formula:a · b = |a||b|cosθwhere θ is the angle between them. So, maybe I can use this formula somewhere here.But wait, the expression I need to compute is (2a - b) · a. That looks like a dot product of two vectors: (2a - b) and a. I think I can expand this expression using the distributive property of the dot product. Let me recall, the dot product is distributive over vector addition, right? So, (u + v) · w = u · w + v · w. Similarly, scalar multiplication can be factored out: (c u) · v = c (u · v), where c is a scalar.So, applying this to (2a - b) · a, I can write it as:2a · a - b · aOkay, that seems manageable. Now, let's compute each term separately.First term: 2a · a. Since a · a is just the magnitude of a squared, right? Because a · a = |a|². So, that would be 2 times |a|².Given that |a| = 2, so |a|² = 2² = 4. Therefore, 2a · a = 2 * 4 = 8.Second term: -b · a. Hmm, that's the negative of the dot product of b and a. But I know that the dot product is commutative, so a · b = b · a. So, -b · a is the same as - (a · b). Now, I need to compute a · b. As I mentioned earlier, a · b = |a||b|cosθ. We have |a| = 2, |b| = 5, and θ = 120 degrees.Let me compute that. So, |a||b| = 2 * 5 = 10. Then, cos(120 degrees). Hmm, what's the cosine of 120 degrees? I remember that 120 degrees is in the second quadrant, and cosine is negative there. Also, 120 degrees is equal to 180 - 60 degrees, so cos(120°) = -cos(60°). Since cos(60°) is 0.5, cos(120°) is -0.5.So, a · b = 10 * (-0.5) = -5.Therefore, - (a · b) = - (-5) = 5.So, putting it all together, the expression (2a - b) · a is equal to 8 + 5 = 13.Wait, hold on, let me double-check that. So, 2a · a is 8, and -b · a is 5, so 8 + 5 is indeed 13. That seems right.But just to make sure I didn't make any mistakes, let me go through the steps again.1. Expand (2a - b) · a: 2a · a - b · a.2. Compute 2a · a: 2 * |a|² = 2 * 4 = 8.3. Compute -b · a: - (a · b) = - (|a||b|cosθ) = - (2 * 5 * cos120°) = - (10 * (-0.5)) = - (-5) = 5.4. Add them together: 8 + 5 = 13.Yes, that seems consistent. So, I think the value is 13.Alternatively, I can think about it geometrically. The expression (2a - b) · a can be interpreted as the projection of (2a - b) onto a multiplied by the magnitude of a. But I think the algebraic approach I took earlier is more straightforward here.Just to explore another angle, if I were to compute (2a - b) · a, I can also think of it as 2(a · a) - (b · a). Which is exactly what I did. So, that reinforces my earlier calculation.I don't think I made any calculation errors. Let me verify the cosine part again. 120 degrees, cosine is -0.5. Yes, that's correct because cos(60°) is 0.5, and since 120° is in the second quadrant, cosine is negative. So, that part is right.Also, |a| is 2, so |a| squared is 4, which multiplied by 2 gives 8. That seems correct.And then, the dot product a · b is 2 * 5 * (-0.5) = -5, so negative of that is 5. So, 8 + 5 is 13.I think that's solid. I don't see any mistakes in my reasoning.**Final Answer**The value of (2 boldsymbol{a} - boldsymbol{b}) cdot boldsymbol{a} is boxed{13}.

answer：Alright, so I have this sequence {a_n} with some specific rules, and I need to find the value of a_{2009} + a_{2014}. Let me try to understand the problem step by step.First, the sequence is defined by three rules:1. a_{4n - 3} = 12. a_{4n - 1} = 03. a_{2n} = a_nWhere n is a positive integer (n ∈ ℕ^*). So, these rules define the sequence for all positive integers n.I need to figure out what a_{2009} and a_{2014} are, then add them together. Let me tackle each term one by one.Starting with a_{2009}.Hmm, 2009 is an odd number. Let me check if it fits into either of the first two cases: 4n - 3 or 4n - 1.Let me solve for n in both cases.Case 1: 4n - 3 = 2009So, 4n = 2009 + 3 = 2012n = 2012 / 4 = 503Since 503 is an integer, that means 2009 is of the form 4n - 3, so according to the first rule, a_{2009} = 1.Okay, so a_{2009} is 1. That was straightforward.Now, moving on to a_{2014}.2014 is an even number, so maybe I can use the third rule: a_{2n} = a_n. Let me apply that.First, write 2014 as 2 * 1007. So, a_{2014} = a_{1007}.Now, 1007 is odd. Let me see if it fits into either 4n - 3 or 4n - 1.Case 1: 4n - 3 = 10074n = 1007 + 3 = 1010n = 1010 / 4 = 252.5Hmm, that's not an integer, so it doesn't fit into the first case.Case 2: 4n - 1 = 10074n = 1007 + 1 = 1008n = 1008 / 4 = 252That's an integer, so 1007 is of the form 4n - 1, which means a_{1007} = 0.Therefore, a_{2014} = a_{1007} = 0.Wait, so a_{2014} is 0.Therefore, adding them together: a_{2009} + a_{2014} = 1 + 0 = 1.But hold on, let me double-check my reasoning because sometimes these recursive sequences can have tricky parts.Starting with a_{2009}:2009 is odd, so I checked if it's 4n - 3 or 4n - 1. It turned out to be 4*503 - 3, which is correct. So a_{2009} = 1. That seems solid.For a_{2014}, since it's even, I reduced it to a_{1007}. 1007 is odd, so I checked the two cases again. It didn't fit into 4n - 3 because n was 252.5, which isn't an integer, but it did fit into 4n - 1 with n = 252. So a_{1007} = 0, hence a_{2014} = 0.Therefore, adding them gives 1 + 0 = 1.Is there another way to think about this? Maybe by looking at the binary representations or something else?Wait, another approach could be to see if these terms correspond to specific positions in the sequence defined by the rules. Maybe the sequence is related to binary digits or something like that.But let me think again. The rules are:1. For terms of the form 4n - 3, which are numbers like 1, 5, 9, 13, ..., the value is 1.2. For terms of the form 4n - 1, which are numbers like 3, 7, 11, 15, ..., the value is 0.3. For even terms, it's equal to the term at half the index.So, this seems similar to a recursive sequence where even indices depend on smaller indices, and odd indices are determined by whether they're 1 mod 4 or 3 mod 4.So, for any number, if it's even, we can keep dividing by 2 until we get an odd number, and then check if that odd number is 1 mod 4 or 3 mod 4.So, for a_{2009}: 2009 is odd. 2009 divided by 4 is 502.25, so 2009 = 4*502 + 1? Wait, 4*502 = 2008, so 2009 = 2008 + 1, which is 4*502 + 1. So, 2009 is 1 mod 4. But according to the first rule, a_{4n - 3} = 1. Let me see: 4n - 3 = 1 when n=1, which is 1. So, 1 is 1 mod 4. Similarly, 5 is 1 mod 4, 9 is 1 mod 4, etc. So, numbers that are 1 mod 4 correspond to a_n = 1.Similarly, numbers that are 3 mod 4 correspond to a_n = 0.So, perhaps the sequence is defined such that a_n is 1 if n is 1 mod 4, and 0 if n is 3 mod 4, and for even numbers, it's equal to a_{n/2}.So, in that case, to compute a_n, we can keep dividing n by 2 until we get an odd number, and then check if that odd number is 1 mod 4 or 3 mod 4.So, for a_{2009}, since it's odd, we check 2009 mod 4. 2009 divided by 4 is 502 with a remainder of 1. So, 2009 ≡ 1 mod 4, so a_{2009} = 1.For a_{2014}, it's even, so we divide by 2 to get 1007. 1007 is odd. 1007 divided by 4 is 251 with a remainder of 3. So, 1007 ≡ 3 mod 4, so a_{1007} = 0, hence a_{2014} = 0.Therefore, the sum is 1 + 0 = 1.Wait, so this seems consistent with my earlier reasoning.Alternatively, maybe I can think of the sequence in terms of binary representations. Let me see.If I write n in binary, perhaps the rules correspond to certain bits. For example, the evenness corresponds to the least significant bit, and dividing by 2 corresponds to shifting right.But maybe that's complicating things. Since the rules are straightforward, perhaps my initial approach is sufficient.Another way to think about it is to model the sequence as a function that depends on the binary representation of n, but I don't see an immediate connection. Maybe it's not necessary here.Alternatively, perhaps I can write a small table for the sequence to see the pattern.Let me try writing out the sequence for the first few terms.n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,...a_n: ?Let's compute them step by step.n=1: It's 4*1 - 3 = 1, so a_1 = 1.n=2: Even, so a_2 = a_1 = 1.n=3: 4*1 - 1 = 3, so a_3 = 0.n=4: Even, a_4 = a_2 = 1.n=5: 4*2 - 3 = 5, so a_5 = 1.n=6: Even, a_6 = a_3 = 0.n=7: 4*2 - 1 = 7, so a_7 = 0.n=8: Even, a_8 = a_4 = 1.n=9: 4*3 - 3 = 9, so a_9 = 1.n=10: Even, a_10 = a_5 = 1.n=11: 4*3 - 1 = 11, so a_11 = 0.n=12: Even, a_12 = a_6 = 0.n=13: 4*4 - 3 = 13, so a_13 = 1.n=14: Even, a_14 = a_7 = 0.n=15: 4*4 - 1 = 15, so a_15 = 0.n=16: Even, a_16 = a_8 = 1.So, the sequence starts as: 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1,...Looking at this, it seems like the sequence is 1 for numbers that are 1 mod 4 or 2 mod 4, and 0 otherwise? Wait, let me check.Wait, n=1: 1 mod 4, a_n=1.n=2: 2 mod 4, a_n=1.n=3: 3 mod 4, a_n=0.n=4: 0 mod 4, a_n=1.n=5: 1 mod 4, a_n=1.n=6: 2 mod 4, a_n=0.Wait, so n=6 is 2 mod 4, but a_n=0.Hmm, so that idea doesn't hold.Wait, n=2: 2 mod 4, a_n=1.n=6: 2 mod 4, a_n=0.So, it's inconsistent.Wait, maybe it's based on the highest power of 2 dividing n?Wait, n=1: 2^0, a_n=1.n=2: 2^1, a_n=1.n=3: 2^0, a_n=0.n=4: 2^2, a_n=1.n=5: 2^0, a_n=1.n=6: 2^1, a_n=0.n=7: 2^0, a_n=0.n=8: 2^3, a_n=1.n=9: 2^0, a_n=1.n=10: 2^1, a_n=1.n=11: 2^0, a_n=0.n=12: 2^2, a_n=0.n=13: 2^0, a_n=1.n=14: 2^1, a_n=0.n=15: 2^0, a_n=0.n=16: 2^4, a_n=1.Hmm, so for numbers where the highest power of 2 dividing n is even? Wait, n=2: 2^1, a_n=1.n=4: 2^2, a_n=1.n=8: 2^3, a_n=1.n=16: 2^4, a_n=1.So, for n being powers of 2, regardless of the exponent, a_n=1.For numbers that are twice a power of 2, like n=2, 6, 10, 14,... Let's see:n=2: a_n=1.n=6: a_n=0.n=10: a_n=1.n=14: a_n=0.Hmm, alternating 1, 0, 1, 0,...Wait, n=2: 2=2*1, a_n=1.n=6=2*3, a_n=0.n=10=2*5, a_n=1.n=14=2*7, a_n=0.So, it alternates based on whether the odd part is 1 mod 4 or 3 mod 4.Wait, n=2: 2=2*1, 1 is 1 mod 4, so a_n=1.n=6=2*3, 3 is 3 mod 4, so a_n=0.n=10=2*5, 5 is 1 mod 4, so a_n=1.n=14=2*7, 7 is 3 mod 4, so a_n=0.So, in general, for even numbers, a_n = a_{n/2}, which depends on whether n/2 is 1 mod 4 or 3 mod 4, or whether n/2 is even, in which case we keep dividing by 2 until we get an odd number.So, the key is that for any n, we can write it as n = 2^k * m, where m is odd. Then, a_n = a_m.And since m is odd, it's either 1 mod 4 or 3 mod 4. If m ≡ 1 mod 4, then a_n = 1; if m ≡ 3 mod 4, then a_n = 0.So, in other words, a_n is 1 if the odd part of n is 1 mod 4, and 0 if the odd part is 3 mod 4.Therefore, to compute a_n, we can factor out all the powers of 2 from n, get the odd part m, and then check m mod 4.If m ≡ 1 mod 4, a_n=1; else, a_n=0.So, applying this to a_{2009}:2009 is odd, so m=2009. 2009 mod 4 is 1, so a_{2009}=1.For a_{2014}:2014 is even. Let's factor out the powers of 2.2014 divided by 2 is 1007, which is odd. So, m=1007. 1007 mod 4 is 3, so a_{2014}=0.Therefore, the sum is 1 + 0 = 1.This seems consistent with my earlier reasoning.Let me test this with another number to make sure.Take n=12. 12 is even. 12 / 2 = 6, which is even. 6 / 2 = 3, which is odd. So, m=3. 3 mod 4 is 3, so a_{12}=0. Which matches the earlier sequence.Similarly, n=10: 10 is even. 10 / 2 = 5, which is odd. 5 mod 4 is 1, so a_{10}=1. Which also matches.n=14: 14 / 2 =7, which is odd. 7 mod 4 is 3, so a_{14}=0. Correct.n=16: 16 is 2^4, so m=1. 1 mod 4 is 1, so a_{16}=1. Correct.Okay, so this method seems reliable.Therefore, applying it to 2009 and 2014:- 2009 is odd, m=2009. 2009 / 4 = 502.25, so 2009 = 4*502 + 1, so 2009 ≡ 1 mod 4. Thus, a_{2009}=1.- 2014 is even. 2014 / 2 = 1007, which is odd. 1007 / 4 = 251.75, so 1007 = 4*251 + 3, so 1007 ≡ 3 mod 4. Thus, a_{2014}=0.Therefore, a_{2009} + a_{2014} = 1 + 0 = 1.I think this is solid. I don't see any mistakes in my reasoning. I've checked it multiple ways: directly applying the rules, considering the sequence's pattern, and breaking down the number into its odd part. All methods lead to the same conclusion.**Final Answer**boxed{1}

answer：First, I note that the ratio of girls to boys is 3:2. This means for every 3 girls, there are 2 boys.Next, I determine the total number of parts in the ratio by adding 3 and 2, which gives me 5 parts.Since there are 45 students in total, each part represents 45 divided by 5, which is 9 students.Finally, to find the number of girls, I multiply the number of parts for girls (3) by the number of students per part (9), resulting in 27 girls.

answer：To find the largest three-digit number where the number minus the sum of its digits is a perfect square, I'll start by considering the largest three-digit number, which is 999.First, I'll calculate the sum of the digits of 999:9 + 9 + 9 = 27Next, I'll subtract this sum from the number:999 - 27 = 972I need to check if 972 is a perfect square. The square of 31 is 961, and the square of 32 is 1024. Since 972 falls between these two values, it's not a perfect square.I'll then decrease the number by 1 and repeat the process:998 - (9 + 9 + 8) = 998 - 26 = 972Again, 972 is not a perfect square. I'll continue this process until I find a number where the result is a perfect square.After several iterations, I find that:990 - (9 + 9 + 0) = 990 - 18 = 972989 - (9 + 8 + 9) = 989 - 26 = 963...961 - (9 + 6 + 1) = 961 - 16 = 945960 - (9 + 6 + 0) = 960 - 15 = 945Finally, I reach 961, which is a perfect square (31²). Therefore, the largest three-digit number that satisfies the condition is 961.

answer：Okay, so I have this problem here: If ((1 - 2x)^{2014} = a_0 + a_1x + a_2x^2 + ldots + a_{2014}x^{2014}) for real numbers (x), then I need to find the value of (a_0 + frac{1}{2}a_1 + frac{1}{2^2}a_2 + frac{1}{2^3}a_3 + ldots + frac{1}{2^{2014}}a_{2014}).Hmm, let me try to figure this out step by step. I remember that when dealing with polynomial expansions, sometimes substituting specific values for (x) can help find the sum of coefficients or other related expressions. The given expression is the expansion of ((1 - 2x)^{2014}), which is a binomial expansion. The coefficients (a_0, a_1, a_2, ldots, a_{2014}) correspond to the terms in this expansion. Now, the problem is asking for the sum (a_0 + frac{1}{2}a_1 + frac{1}{2^2}a_2 + ldots + frac{1}{2^{2014}}a_{2014}). That looks like evaluating the polynomial at a specific value of (x), but not just (x = 1) or (x = 0). Instead, each term is scaled by a factor of (frac{1}{2^k}) where (k) is the degree of the term. Wait, that sounds familiar. If I substitute (x = frac{1}{2}) into the polynomial, wouldn't that give me exactly the sum (a_0 + frac{1}{2}a_1 + frac{1}{2^2}a_2 + ldots + frac{1}{2^{2014}}a_{2014})? Let me check:If I plug (x = frac{1}{2}) into ((1 - 2x)^{2014}), I get:[(1 - 2 times frac{1}{2})^{2014} = (1 - 1)^{2014} = 0^{2014} = 0]But wait, that would mean the sum is zero? That seems too straightforward. Let me verify if I'm interpreting the substitution correctly.Yes, substituting (x = frac{1}{2}) into the expansion gives:[a_0 + a_1 times frac{1}{2} + a_2 times left(frac{1}{2}right)^2 + ldots + a_{2014} times left(frac{1}{2}right)^{2014}]Which is exactly the sum we're asked to compute. So, according to this substitution, the sum is zero. But let me think again—does this make sense? The expansion of ((1 - 2x)^{2014}) when (x = frac{1}{2}) is indeed zero, because (1 - 2 times frac{1}{2} = 0). So, the entire expression becomes zero. Is there another way to approach this problem? Maybe using generating functions or another substitution?Alternatively, I can think about the general form of the binomial expansion. The expansion of ((1 - 2x)^{2014}) is:[sum_{k=0}^{2014} binom{2014}{k} (1)^{2014 - k} (-2x)^k = sum_{k=0}^{2014} binom{2014}{k} (-2)^k x^k]So, the coefficients (a_k) are given by (binom{2014}{k} (-2)^k). Therefore, the sum we need is:[sum_{k=0}^{2014} frac{1}{2^k} a_k = sum_{k=0}^{2014} frac{1}{2^k} times binom{2014}{k} (-2)^k]Simplifying the terms inside the sum:[frac{1}{2^k} times (-2)^k = left( frac{-2}{2} right)^k = (-1)^k]So, the sum becomes:[sum_{k=0}^{2014} binom{2014}{k} (-1)^k]Wait a second, that's the expansion of ((1 - 1)^{2014}), which is indeed (0^{2014} = 0). So, this confirms the earlier result.Therefore, both methods lead to the conclusion that the sum is zero. But just to ensure I haven't missed anything, let me consider the substitution method again. If I set (x = frac{1}{2}), the original polynomial becomes zero, which directly gives the sum as zero. So, both substitution and expanding the coefficients lead to the same answer.I think I'm confident now that the value is zero.**Final Answer**boxed{0}