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  4. The terms a1, a2, a3 (in that order) form an increasing arithmetic sequence whose sum is 18 . If a 1+1, a 2, a 3+2 form a geometric sequence (in that order), then sum of all possible values of a 1 is

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Q. The terms $a_1, a_2, a_3$ (in that order) form an increasing arithmetic sequence whose sum is 18 . If $a _1+1, a _2, a _3+2$ form a geometric sequence (in that order), then sum of all possible values of $a _1$ is

Sequences and Series

Solution:

$a-d, a, a+d \rightarrow$ A.P.
$3 a =18 $
$\therefore a =6$
$\text { Now } 7- d , 6,8+ d \rightarrow \text { G.P. }$
$\therefore 36=(7- d )(8+ d ) $
$\therefore d =4 \text { or }-5 \text { (rejected) } $
$\Rightarrow a _1=2 \Rightarrow S =2 .$