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10. In the expansion of (2 + 3x)", the coefficient of x' and x are in the ratio 8: 15. Find the value of n. ​​​

User Laurena
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Final answer:

The question concerns finding the value of n using the ratio of coefficients in the binomial expansion (2 + 3x)^n. The binomial theorem is applied to determine the coefficients of x^2 and x, and the ratio is set up as 8:15 to solve for n.

Step-by-step explanation:

The student's question involves finding the value of n in the expansion of (2 + 3x)^n, where the coefficient of x^2 and x are in the ratio 8:15. To solve this problem, we use the binomial theorem, which states that the expansion of (a + b)^n is a^n + na^{n-1}b + n(n-1)/2! a^{n-2}b^2 + ... and so on.

In the expansion of (2 + 3x)^n, the coefficient of x^2 is n(n-1)/2! × 2^{n-2} × 3^2 and the coefficient of x is n × 2^{n-1} × 3. We set up the ratio of these coefficients as 8/15 and solve for n.

User Hhbcarl
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