Q13 of 97 Page 199

If , then show that a, b, c and d are in G.P.

It is given that,

On cross multiplying, we get -

(a + bx)(b - cx) = (b + cx)(a - bx)

⇒ ab - acx + b²x - bcx² = ab - b²x + acx - bcx²

⇒ 2b²x = 2acx

⇒ b² = ac

…(1)

Also,

On cross multiplying, we get -

⇒ (b + cx)(c - dx) = (c + dx)(b - cx)

⇒ bc - bdx + c²x - cdx² = bc + bdx - c²x - cdx²

⇒ 2c²x = 2bdx

⇒ c² = bd

…(2)

From (1) and (2), we obtain

Thus, a, b, c, and d are in G.P.

A G.P. consists of an even number of terms. If the sum of all the terms is 6 times the sum of terms occupying odd places, then find its common ratio.

The sum of the first four terms of an A.P. is 66. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P.

Prove that P²Rⁿ = Sⁿ.

The p^th, q^th and r^th terms of an A.P. are a, b, c, respectively. Show that

(q – r )a + (r – p )b + (p – q )c = 0