Let the first three terms 2, p and q, with q ≠ 2, of a G.P. be respectively the 7th, 8th and 13th terms

Question

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 \(\mathrm{n}^{\text {th }}\) term of the A.P., then \(n\) is equal to

(1) 151

(2) 169

(3) 177

(4) 163

Answer 1

Correct option is (4) 163

\(p^{2}=2 q\)

\(2=a+6 d \quad \ldots(i)\)

\(\mathrm{q}=\mathrm{a}+12 \mathrm{~d} \quad ....(ii)\)

\(\mathrm{q}=\mathrm{a}+12 \mathrm{~d} \quad ....(iii)\)

\(\mathrm{p}-2=\mathrm{d} \quad ((ii) - (i))\)

\(\mathrm{q}-\mathrm{p}=5 \mathrm{~d} \quad((ii) - (i))\) 

\(q-p=5(p-2)\)

\(q=6 p-10\)

\(\mathrm{p}^{2}=2(6 \mathrm{p}-10)\)

\(\mathrm{p}^{2}-12 \mathrm{p}+20=0\)

\(\mathrm{p}=10,2\)

\(p=10 ; q=50\)

\(\mathrm{d}=8\)

\(a=-46\)

\(2, 10, 50, 250, 1250\)

\(a r^{4}=a+(n-1) d\)

\(1250=-46+(n-1) 8\)

\(\mathrm{n}=163\)