Class 11

Math

Algebra

Sequences and Series

If a, b, c are in A.P., b, c, d are in G.P. and $c1 ,d1 ,e1 $are in A.P. prove that a, c, e are in G.P.

$a−b=b−c$

$ba −1=1−bc →(1)$

As, b,c and d are in G.P.

$bd=c_{2}→(2)$

As, 1/c,1/d,1/e are in A.P.

$d1 −e1 =e1 −d1 $

$1−cd =ed −1→(3)$

From (2)$1−bc =ed −1$

From (1)$ba −1=ed −1$

$ae=bd$

As, $bd=c_{2},$$ae=c_{2}$,Thus, a,c and e are in G.P.