Question

Xn is the n-th term of an Arithmetic Sequence. If Xa, Xb, Xc...... are in arithmetic sequence, prove that a, b, c are in Arithmetic Sequence ​

Answer 1

Don't get confused how to start this question. Simply run ur mind in 2b = a + c direction. Let's solve it step - by step!

Solution: If Xa, Xb and Xc are in A.P. then,

→ 2Xb = Xa + Xc

Now, we know general Formula of A.P.,

• Xa = X + (a - 1)d
• Xb = X + (b - 1)d
• Xc = X + (c - 1)d

→ 2[X + (b - 1)d] = [X + (a - 1)d] + [X + (c - 1)d]

→ 2X + 2d(b - 1) = 2X + d[(a - 1) + (c - 1)]

→ 2d(b - 1) = d(a + c - 2)

→ 2(b - 1) = (a + c - 2)

→ 2b - 2 = a + c - 2

→ 2b = a + c

Hence, a, b and c are in A.P.

This wasn't difficult ryt? Happy Learning! :)