Algebra
# Geometric Progressions

Evaluate

\[ 1 + 2 + 2^2 + 2^3 + \ldots + 2^{ 24 }. \]

A bookshelf has \(6\) shelves. The top shelf has \(2\) books, and each other shelf has \(3\) times as many books as the shelf directly above it. If you want to read a book from the shelves, how many choices do you have?

**Details and assumptions**

All of the books are different.

The series \(a_1, a_2, a_3, a_4, a_5, a_6, a_7\) is a geometric progression with \(a_4 = 48\) and \(a_5 = 96\). What is the sum of the these \(7\) terms?

**Details and assumptions**

A **geometric progression** is a sequence of numbers, in which each subsequent number is obtained by multiplying the previous number by a common ratio / multiple. For example, \(1, 2, 4, 8, 16, 32, 64, \ldots \) is a geometric progression with initial term 1 and common ratio 2.

Evaluate the sum to infinity:

\[ \frac{ 14 } { 2^0} + \frac{ 14 } { 2^1 } + \frac{ 14 } { 2^2 } + \ldots .\]