Geometric Series

Starter: Describe the recurrence relation for each sequence Geometric Series KUS objectives BAT you need to be able to spot patterns to work out the common ratio and rule for a Geometric Sequence BAT use Percentages in Geometric Sequences BAT apply logarithms to solve problems Starter: Describe the recurrence relation for each sequence (formula for the next term) 3, 12, 48, 192, … 𝑢 𝑛+1 =4× 𝑢 𝑛 4, 2, 1, 1 2 , 1 4 , 1 8 , … 𝑢 𝑛+1 = 1 2 × 𝑢 𝑛 6× 2 5 , 6× 4 25 , 6× 8 125 , … 𝑢 𝑛+1 = 2 5 × 𝑢 𝑛

r is the common ratio, and is the same for all consecutive pairs WB1 Common ratio In a Geometric sequence, u1, u2, u3 … un we will have 𝑈𝑛 𝑈 𝑛+1 =𝒓 a constant value r is the common ratio, and is the same for all consecutive pairs Find the common ratio in each of these sequences: i) 4, 2, 1, 1 2 , 1 4 , 1 8 , … 𝐫= 2 4 = 1 2 𝐫= 192 48 = 3 ii) 3, 12, 48, 192, … 𝐫= 18 54 = 1 3 iii) 54, 18, 6, 2, … 𝐫= 1 10 iv) 21, 2.1, 0.21, 0.021, … ⟹ 𝑥 2 =1 ⟹ 𝑥 =±4 ⟹ 𝒓= 𝑥 20 =0.2 𝐫= 0.8 𝑥 = 𝑥 20 v) 20, 𝑥, 0.8, … 𝐫= 51.2 𝑥 = 𝑥 5 ⟹ 𝑥 2 =256 ⟹ 𝑥 =±16 ⟹ 𝒓= 𝑥 5 =3.2 vi) 5, 𝑥, 51.2, …

… ,4× 1 2 𝑛_1 … ,3× 4 𝑛_1 … ,54× 1 3 𝑛_1 ,… ,21× 1 10 𝑛_1 WB2 Nth term In a Geometric sequence, the nth term comes from multiplying the first term by the common ration (n – 1) times a, ar, ar2, ar3, …, arn-1 1st Term 2nd Term 3rd Term 4th Term nth Term For example. find the nth term of these: … ,4× 1 2 𝑛_1 i) 4, 2, 1, 1 2 , 1 4 , 1 8 , … ii) 3, 12, 48, 192,… … ,3× 4 𝑛_1 … ,54× 1 3 𝑛_1 iii) 54, 18, 6, 2,… ,… ,21× 1 10 𝑛_1 iv) 21, 2.1, 0.21, 0.021, …

Find the nth and 10th terms of the following sequences… WB 3 nth term Find the nth and 10th terms of the following sequences… i) 40, −20, 10, −5, … 𝑎=3, 𝑟=2, 𝑎 𝑟 𝑛−1 =3 × 2 𝑛−1 10𝑡ℎ=1536 ii) 3, 12, 48, 192, … 𝑎=40, 𝑟=− 1 2 , 𝑎 𝑟 𝑛−1 =40 × 1 2 𝑛−1 10𝑡ℎ=− 5 64 iii) 2, 8, 32, … iv) 12, −3, 3 4 , … v) 1 5 , 1 50 , 1 500 , … vi) 6× 2 5 , 6× 4 25 , 6× 8 125 , …

WB 4 Finding a and r The second term of a Geometric sequence is 4, and the 4th term is 8. Find the values of the common ratio and the first term 1 2nd Term  2 4th Term  2 ÷ 1  Square root Sub r into 1 Divide by √2 Rationalise

a) A geometric sequence has third term 12 and sixth term -96 WB 5ab finding a and r a) A geometric sequence has third term 12 and sixth term -96 Find the first term and the common ratio 𝑢 6 𝑢 3 = 𝑎 𝑟 5 𝑎 𝑟 2 = 𝑟 3 𝑢 6 𝑢 3 = −96 12 =−8 So 𝑟=−2 So third term 𝑎 𝑟 2 =4𝑎=12 So first term 𝑎=3 b) The third term of a geometric sequence is 324 and the fifth term is 36 Find the first term and the two possible values of the common ratio 𝑢 5 𝑢 3 = 𝑎 𝑟 4 𝑎 𝑟 2 = 𝑟 2 𝑢 5 𝑢 3 = 36 324 = 1 9 So 𝑟= 1 3 So third term 𝑎 𝑟 2 = 1 9 𝑎=324 So first term 𝑎=2916

WB 5c finding a and r c) The numbers 3, x, and (x + 6) form the first three terms of a positive geometric sequence. Calculate the 15th term of the sequence First term = 3 Common Ratio = 2 Nth term = 3 x 2n-1 15th Term = 3 x 214 x has to be positive 15th Term = 49152

So first term when 3 𝑛 >10000 WB 6 using logarithms Find the first term in the geometric sequence 1, 3, 9, 27 … to exceed the value of 10000 𝑟=3 𝑎=1 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 3 𝑛 So first term when 3 𝑛 >10000 use logarithms n> 𝑙𝑜𝑔 3 10000 So n> 8.38 first term that works is n = 9

WB7: logarithms What is the first term in the sequence 3, 6, 12, 24… to exceed 1 million? 𝑟=2 𝑎=3 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 3× 2 𝑛−1 So first term when 2 𝑛−1 > 1000000 3 use logarithms n−1 > 𝑙𝑜𝑔 2 1000000 3 =18.35 So n>19.35 first term that works is n = 20 b) Find the first term in the geometric sequence 2, 12, 72, 432, … to exceed the value of 400000 𝑟=6 𝑎=2 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 2×6 𝑛−1 So first term when 2× 6 𝑛−1 > 400000 use logarithms n−1> 𝑙𝑜𝑔 6 400000 2 =6.81 So n> 7.81 first term that works is n = 8

Using percentage change If I was to increase an amount by 10%, what would I multiply the value by? If I was to increase an amount by 17%, what would I multiply by?  1.1  1.17 If I was to increase an amount by 10%, every year for 6 years, what would I multiply the value by? If I was to increase an amount by 17%, every year for 20 years, what would I multiply by?  × 𝟏.𝟏 𝟔  × 𝟏.𝟏𝟕 𝟐𝟎

If £A is to be invested in a savings fund at a rate of 4%. WB 8 percentage change If £A is to be invested in a savings fund at a rate of 4%. How much should be invested so the fund is worth £10,000 in 5 years? Y1 Y2 Y3 Y4 Y5 A Ar Ar5 Ar4 Ar3 Ar2 a = A r = 1.04 Ar5 = 10,000 A x 1.045 = 10,000 A = £8219.27

One thing to improve is – KUS objectives BAT you need to be able to spot patterns to work out the common ratio and rule for a Geometric Sequence BAT use Percentages in Geometric Sequences BAT apply logarithms to solve problems self-assess One thing learned is – One thing to improve is –

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