Geometric Sequences and Series Summation

The sequence is an example of a Geometric sequence A sequence is geometric if where r is a constant called the common ratio In the above sequence, r = 2

A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is

Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences (a) Ans: (b) Ans: (c) Ans:

The formula will be proved next but you don’t need to learn the proof. Summing terms of a G.P. e.g.1 Evaluate Writing out the terms helps us to recognize the G.P. With a calculator we can see that the sum is 186. But we need a formula that can be used for any G.P. The formula will be proved next but you don’t need to learn the proof.

Summing terms of a G.P. With 5 terms of the general G.P., we have TRICK Multiply by r: Subtracting the expressions gives Move the lower row 1 place to the right

Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives and subtract

Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives

Summing terms of a G.P. So, Take out the common factors and divide by ( 1 – r ) Similarly, for n terms we get

Summing terms of a G.P. gives a negative denominator if r > 1 The formula Instead, we can use

Summing terms of a G.P. For our series Using

Summing terms of a G.P. EX Find the sum of the first 20 terms of the geometric series, leaving your answer in index form Solution: We’ll simplify this answer without using a calculator

Summing terms of a G.P. There are 20 minus signs here and 1 more outside the bracket!

3rd term + 4th term = 4( 1st term + 2nd term ) Summing terms of a G.P. e.g. 3 In a geometric sequence, the sum of the 3rd and 4th terms is 4 times the sum of the 1st and 2nd terms. Given that the common ratio is not –1, find its possible values. Solution: As there are so few terms, we don’t need the formula for a sum 3rd term + 4th term = 4( 1st term + 2nd term ) Divide by a since the 1st term, a, cannot be zero:

Summing terms of a G.P. We need to solve the cubic equation Should use the factor theorem: We will do this soon !!

Summing terms of a G.P. The solution to this cubic equation is therefore Since we were told we get

SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is The sum of n terms is or

Sum to Infinity IF |r|<1 then Because (<1)∞ = 0

2. Find the sum of the first 15 terms of the G.P. Exercises 1. Find the sum of the first 15 terms of the following G.P., giving the answers in index form 2 + 8 + 32 + . . . 2. Find the sum of the first 15 terms of the G.P. 4 - 2 + 1 + . . . giving your answer correct to 3 significant figures.

2 + 8 + 32 + . . . 4 - 2 + 1 + . . . Exercises 1. Solution: 2 + 8 + 32 + . . . 2. Solution: 4 - 2 + 1 + . . . ( 3 s.f. )