Geometric 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.,

wherercommon ratio
 a1first term
 a2second term
 a3third term
 an-1the term before the n th term
 anthe 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:

wherercommon ratio
 a1first term
 an-1the term before the n th term
 nnumber 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

whereSnsum of GP with n terms
 Ssum of GP with infinitely many terms
 a1the first term
 rcommon ratio
 nnumber 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 are repeated in period are called recurring decimals.

For example, 0.22222222... is a recurring decimal because the number 2 is repeated infinitely.

The recurring decimal 0.22222222... can be written as .

Another example is 0.234523452345... is a recurring decimal because the number 2345 is repeated periodically.

Thus, it can be written as or it can also be expressed in fractions.

Question

Express as a fraction in their lowest terms.

Answer

Go to the next page to start putting what you have learnt into practice.

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