Calculus

Sequences and Series

Geometric Progression Sum

         

Evaluate

1+2+22+23++224. 1 + 2 + 2^2 + 2^3 + \ldots + 2^{ 24 }.

A bookshelf has 66 shelves. The top shelf has 22 books, and each other shelf has 33 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 a1,a2,a3,a4,a5,a6,a7a_1, a_2, a_3, a_4, a_5, a_6, a_7 is a geometric progression with a4=48a_4 = 48 and a5=96a_5 = 96. What is the sum of the these 77 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, 1,2,4,8,16,32,64,1, 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 44 and common ratio 55. What is the smallest value of nn for which the sum of the first nn terms is greater than 800800?

Evaluate the sum to infinity:

1420+1421+1422+. \frac{ 14 } { 2^0} + \frac{ 14 } { 2^1 } + \frac{ 14 } { 2^2 } + \ldots .

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