A Staircase Sequence

★ 8/10Arithmetic Progression Geometric Progression Recursion sequences

Problem

Let $\{a_n\}$ be a geometric sequence with positive common ratio, partial sums $S_n$ , $a_1 = 1$ , and $S_2 = a_3 - 1$ .

1. Find $S_n$ . 2. Define a sequence $\{b_n\}$ by

b_n = \begin{cases} k, & n = a_k, \\ b_{n-1} + 2k, & a_k < n < a_{k+1}, \end{cases}

where $k$ ranges over positive integers greater than $1$ .

When $n = a_{k+1}$ , prove that $b_{n-1} \geqslant a_k \cdot b_n$ .
Find $\displaystyle\sum_{i=1}^{S_n} b_i$ .

Answer

Solution

Difficulty	8/10
Topics	Arithmetic Progression, Geometric Progression, Recursion, sequences

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