Geometric Series

Geometric Series KUS objectives BAT understand how a series can converge or diverge BAT work out the sum to infinity of a Geometric Sequence Starter: Sketch the graph of 𝑦= 2 𝑥 describe its shape Sketch the graph of 𝑦= 2 −𝑥 describe its shape Sketch the graph of 𝑦= 1 2 −𝑥 describe its shape

Infinite sum: Activity: Explore the sum of series Your teacher will give you each a sequence to investigate For each sequence work out the sum of n terms by calculation for S5 S10 S15 S20 S25 S30 S35 S40 …… Describe what happens Can you explain why?

Explore the sequence with the following formula: WB1 Infinite sum Explore the sequence with the following formula: This sequence CONVERGES to 0.1 recurring… First 4 terms As Decimals A Sequence will converge if the common ratio, r is between -1 and 1. Sum of 1st term Sum of 1st and 2nd terms If r>1 or r<-1 the sequence will diverge If r= 1 all the terms are the same. If r = -1 the terms are the same but alternate signs Sum of 1st to 3rd terms Sum of 1st to 4th terms What happens to 𝑺∞ if the common ratio is = 1 or > 1 (or < -1)? For example if r = 2 …

If we start with the formula for the sum of a Geometric sequence: WB1 Infinite sum: formula If we start with the formula for the sum of a Geometric sequence: Think about what happens to 𝑟 𝑛 if −1<𝑟<1, and n increases (𝑛→∞) If we have a value like this which we keep increasing the power of, the value becomes increasingly small and tends towards 0… For example, if r = 0.5 and we keep increasing n… 𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟 For the conditions stated to the right, rn will tend towards 0 as the sequence continues to infinity 𝑆 𝑛 = 𝑎 1−0 1−𝑟 0.51  0.5 Simplify 0.52  0.25 𝑆 𝑛 = 𝑎 1−𝑟 0.53  0.125 0.54  0.0625 This formula calculates the sum to infinity of a sequence, if -1 < r < 1 0.510  0.00097…

Find the sum to infinity of the following sequence: WB 2 Find the sum to infinity of the following sequence: 40 + 10 + 2.5 + 0.625… Substitute Work it out!

First term 𝑎=20 𝑟= 2 3 Sum to infinity 𝑎 1−𝑟 = 20 1− 2 3 =60 WB 3 Find the sum to infinity of the geometric series 20 × 2 3 𝑛−1 First term 𝑎=20 𝑟= 2 3 Sum to infinity 𝑎 1−𝑟 = 20 1− 2 3 =60

Sum to infinity 𝑆 ∞ = 16 1−𝑟 =20 So 𝑟= 1 5 WB 4 The sum to infinity of a convergent series is 20. The first term is 16. Find the third term of the series Sum to infinity 𝑆 ∞ = 16 1−𝑟 =20 So 𝑟= 1 5 So the sequence is 16, 16 5 , 16 25 , 16 125 , … Third term 16 25

WB 5 algebra problem The Sum to infinity of a Sequence is 16, and the sum of the first 4 terms is 15. Find the possible values of r, and the first term if all terms are positive… Sub in 15, and n = 4 Replace a Cancel out (1 - r) Divide by 16 Subtract 1 𝑎=8

a) Sum to infinity 𝑆 ∞ = 22 1−0.6 =55 WB 6 exam Q A geometric progression has first term 22 and common ratio 0.6 Find the sum to infinity Find the sum of the first 34 terms Use logarithms to find the smallest value of p such that the pth term is less than 0.3 a) Sum to infinity 𝑆 ∞ = 22 1−0.6 =55 b) 𝑎 = 22, 𝑟=0.6 𝑆 34 = 22 1− (0.6) 34 1−0.6 = 55 c) 𝑎 𝑟 𝑝−1 =22 (0.6) 𝑝−1 <0.3 (0.6) 𝑝−1 <0.0136 𝑝−1 < 𝑙𝑜𝑔 0.6 0.0136 =8.41 So 𝑝=10

One thing to improve is – KUS objectives BAT understand how a series can converge or diverge BAT work out the sum to infinity of a Geometric Sequence self-assess One thing learned is – One thing to improve is –

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