JEE MAIN - Mathematics (2024 - 4th April Morning Shift - No. 14)
Let the first three terms 2, p and q, with $$q \neq 2$$, of a G.P. be respectively the $$7^{\text {th }}, 8^{\text {th }}$$ and $$13^{\text {th }}$$ terms of an A.P. If the $$5^{\text {th }}$$ term of the G.P. is the $$n^{\text {th }}$$ term of the A.P., then $n$ is equal to:
151
177
163
169
Explanation
$$\begin{aligned} & \text { Let } p=2 r, q=2 r^2 \\ & T_7=2, T_8=2 r, T_{13}=2 r^2 \\ & d=2 r-2=2(r-1) \\ & 2 r^2=T_7+6 d=2+6(2)(r-1)=12 r-10 \\ & \Rightarrow r^2-6 r+5=0 \\ & \Rightarrow(r-1)(r-5)=0 \\ & \therefore r=1,5 \\ & r=1 \text { (rejected) as } q \neq 2 \\ & \therefore r=5 \end{aligned}$$
$$5^{\text {th }}$$ term of G.P $$=2 . r^4=2.5^4$$
Let $$1^{\text {st }}$$ term of A.P $$b a=a, d=8$$
$$2=a+(6)(8) \Rightarrow a=-46$$
$$\mathrm{n}^{\text {th }}$$ term of A.P $$=-46+(n-1) 8=8 n-54$$
$$\begin{aligned} & 2.5^4=8 n-54 \\ & \Rightarrow 1250+54=8 n \\ & \Rightarrow n=\frac{1304}{8}=163 \end{aligned}$$
Comments (0)
