Question

If the 17th and the 18th terms in the expansion of $(2 + a)^{50}$ are equal, then the coefficient of $x^{35}$ in the expansion of $(a + x)^{-2}$ is:

• A. $-35$
• B. $3$
• C. $36$
• D. $-36$

Hint:

In problems involving equal terms of a binomial expansion, equating the terms helps solve for unknowns. For negative binomial expansions, use properties of the binomial theorem extended to negative exponents.

Answer 1

The Correct Option is D

Given the terms $T_{17}$ and $T_{18}$ are equal in the expansion $(2 + a)^{50}$: $$T_{17} = T_{18} \implies \binom{50}{16} 2^{34} a^{16} = \binom{50}{17} 2^{33} a^{17}$$ Solving for $a$, we find: $$a = 2$$ Now, to find the coefficient of $x^{35}$ in $(a + x)^{-2}$: $$(a + x)^{-2} = (2 + x)^{-2}$$ The general term for the binomial expansion is given by: $$T_{r+1} = \binom{-2}{r} 2^{-2-r} x^{r}$$ For $r = 35$, the term is: $$T_{36} = \binom{-2}{35} 2^{-37} x^{35} = (-1)^{35} \frac{(-2)(-3) \ldots (-36)}{35!} \cdot 2^{-37}$$ This simplifies to: $$T_{36} = -\binom{36}{35} \cdot 2^{-37} = -36 \cdot 2^{-37}$$ The coefficient of $x^{35}$ is $-36$, matching option (D).