Geometric Progression: Series and Sums

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ionGeometric Progression, Series & Sums
Introduction
A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where| r| common ratio|
 | a1| first term|
 | a2| second term|
 | a3| third term|
 | an-1| the term before the n th term|
 | an| the n th term|
The geometric sequence is sometimes called the geometric progression or GP, for short. For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3. The geometric sequence has its sequence formation: 

To find the nth term of a geometric sequence we use the formula:

where| r| common ratio|
 | a1| first term|
 | an-1| the term before the n th term|
 | n| number of terms|

Sum of Terms in a Geometric Progression
Finding the sum of terms in a geometric progression is easily obtained by applying the formulas: nth partial sum of a geometric sequence

sum to infinity

where| Sn| sum of GP with n terms|
 | S∞| sum of GP with infinitely many terms|
 | a1| the first term|
 | r| common ratio|
 | n| number of terms|

Examples of Common Problems to Solve
Write down a specific term in a Geometric Progression
Question
Write down the 8th term in the Geometric Progression 1, 3, 9, ... Answer

Finding the number of terms in a Geometric Progression
Question
Find the number of terms in the geometric progression 6, 12, 24, ..., 1536 Answer

Finding the sum of a Geometric Series
Question
Find the sum of each of the geometric series| |
Answer

Finding the sum of a Geometric Series to Infinity
Question

Answer

Converting a Recurring Decimal to a Fraction
Decimals that occurs in repetition infinitely or...
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