Suppose is a geometric sequence with common ratio

MATHEMATICS · Subjective · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Suppose u1,u2,…,unu_1, u_2, \dots, u_n is a geometric sequence with common ratio rr. Show that (u1+u2)2+(u2+u3)2+(u3+u4)2+⋯+(un−1+un)2=2u12(r2n−1−1)r−1−(u12+un2).(u_1 + u_2)^2 + (u_2 + u_3)^2 + (u_3 + u_4)^2 + \dots + (u_{n-1} + u_n)^2 = \frac{2u_1^2(r^{2n-1} - 1)}{r - 1} - (u_1^2 + u_n^2).

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