Q. 5 A3.9( 8 Votes )

# If a, b, c are in A.P., prove that:

(a - c)^{2} = 4 (a - b) (b - c)

Answer :

(a - c)^{2} = 4 (a - b) (b - c)

a^{2} + c^{2} - 2ac = 4(ab – ac – b^{2} + bc)

a^{2} + 4c^{2}b^{2} + 2ac - 4ab - 4bc = 0

(a + c - 2b)^{2} = 0

a + c - 2b = 0

Since a, b, c are in AP

b - a = c - b

a + c - 2b = 0

Hence,

(a - c)^{2} = 4 (a - b) (b - c)

Rate this question :

There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.

RD Sharma - MathematicsIf x, y, z are in A.P. and A_{1}is the A.M. of x and y, and A_{2} is the A.M. of y and z, then prove that the A.M. of A_{1} and A_{2} is y.

Insert 7 A.M.s between 2 and 17.

RD Sharma - MathematicsThe 10^{th} and 18^{th} term of an A.P. are 41 and 73 respectively, find 26^{th} term.

If 10 times the 10^{th} term of an A.P. is equal to 15 times the 15^{th} term, show that the 25^{th} term of the A.P. is Zero.

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P

RD Sharma - MathematicsThe 4^{th} term of an A.P. is three times the first and the 7^{th} term exceeds twice the third term by 1. Find the first term and the common difference.

In an A.P. the first term is 2, and the sum of the first 5 terms is one-fourth of the next 5 terms. Show that 20th term is - 112

RD Sharma - MathematicsInsert 4 A.M.s between 4 and 19.

RD Sharma - Mathematics