# Let a,b and c be the 7th,11th and 13th terms respectively of a non-constant A.P.If these are also the three consecutive terms of a G.P., then a/c is equal to :Option 1)  2Option 2)  7/13Option 3)  1/2Option 4)  4

General term of an A.P. -

$T_{n}= a+\left ( n-1 \right )d$

- wherein

$a\rightarrow$ First term

$n\rightarrow$ number of term

$d\rightarrow$ common difference

Selection of terms in G.P. -

If we have to take three terms in GP, we take them as $\frac{a}{r},a,ar$

- wherein

Extension : If we have to take (2K+1) term in GP, we take them as

$\frac{a}{r^{k}},\frac{a}{r^{k-1}},---------\frac{a}{r},a,ar------ar^{k}$

From the concept

$\\a = A + 6d \\b= A + 10d \\c = A + 12d$

and $a, b$ and $c$ are in G.P

$\\\Rightarrow (A+10d)^2 = (A+6d)(A+12d)\\\Rightarrow\frac{A}{d} = -14 \\ \frac{a}{c} = \frac{A+6d}{A+12d} = \frac{6 + \frac{A}{d}}{12 + \frac{A}{d}} = \frac{6-14}{12 -14} = 4$

Option 1)

2

Option 2)

7/13

Option 3)

1/2

Option 4)

4

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