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.
Consider a geometric progression with initial term \(4\) and common ratio \(5\). What is the smallest value of \(n\) for which the sum of the first \(n\) terms is greater than \(800\)?
Evaluate the sum to infinity:
\[ \frac{ 14 } { 2^0} + \frac{ 14 } { 2^1 } + \frac{ 14 } { 2^2 } + \ldots .\]