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  4. A sequence of three real numbers form an A.P. and the first term is 9. If 2 is added to the second term and 20 is added to the third term, the resulting numbers will form a G.P. The least value of the third term of the G.P. is

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Q. A sequence of three real numbers form an A.P. and the first term is 9. If 2 is added to the second term and 20 is added to the third term, the resulting numbers will form a G.P. The least value of the third term of the G.P. is

Sequences and Series

Solution:

$9,9+ d , 9+2 d \text { in A.P. }$
Also $9,11+ d , 29+2 d$ in G.P.
Hence $(11+d)^2=9(29+2 d)$
Simplifying$d ^2+4 d -140=0$
$(d+14)(d-10)=0$
$d=-14$ or $d=10 $
$\Rightarrow$ possible A.P.'s are $9,-5,-19 \ldots$ or $9,19,29$
Possible G.P. is $9,-3,1$ or $9,21,49$ with common ratio $-1 / 3$ or $7 / 3$
Hence least $3^{\text {rd }}$ term is 1