example of problem solving in geometric sequence with solution and answer
Answers
Find the 10 th term of a geometric sequence if a1 = 45 and the common ration r = 0.2. Solution Use the formula an = a1 * rn-1 that gives the n th term to find a10 as follows
a10 = a1 * rn-1
= 45 * 0.29 = 2.304 * 10-5
Geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
Geometric sequence
A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
a , ar , ar² , ... , arⁿ⁻¹
The nth term of a geometric sequence with the first term a and the common ratio r is given by: aₙ = arⁿ⁻¹
Sum is given by Sₙ = a(rⁿ - 1)/(r - 1)
Sum of infinite series is given by a/(1 - r) where -1 < r < 1
Geometric Series
The sum of the terms of a geometric sequence is called a geometric series.
Finite geometric sequence will have fixed number of terms.
1 2 4 8 16
This is an GP with first term = 1
common Ratio = 2
Number of Terms = 5
5 15 45 135
This is an GP with first term = 5
common Ratio = 3
Number of Terms = 4
3 -6 12 - 24 48 -96
This is an GP with first term = 3
common Ratio = -2
Number of Terms = 6
Infinite geometric sequence will have infinite terms
1 1/2 1/4 1/8 1/16 , ...
This is an GP with first term = 1
common Ratio = 1/2
Number of Terms = Infinite
2 -1 1/2 -1/4 1/8 -1/16 ..
This is an GP with first term = 2
common Ratio = -1/2
Number of Terms = Infinite
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