Question

Question 20 If a, b, c are in A.P,; b, c, d are in G.P and 1/c, 1/d, 1/e are in A.P. prove that a, c, e are in G.P.

Class X1 - Maths -Sequences and Series Page 200

Answer 1

a , b , c are in AP.
so, common difference is always same
b - a = c - b
2b = a + c
b = (a + c)/2 ------------(1)

again, b , c , d are in GP.
so, common ratio is always same.
c/b = d/c
c² = bd ----------------(2)

also, 1/c , 1/d , 1/e are in AP.
2/d = (1/c + 1/e )
d = 2ce/( c + e ) ---------------(3)

now, put the values of b and d from equations (1) and (3) , in equation (2) .
c² = bd
= { (a + c)/2} × {2ce/(c + e)}
= ce(a + c)/(c + e )
c²(c + e ) = ce(a + c )
c( c + e ) = e(a + c )
c² + ce = ae + ce
c² = ae
c/a = e/c = common ratio is always same .
so, a , c , e are in GP .