Geometric Sequences and Series Part III Geometric Sequences and Series The sequence 1, 2, 4, 8, . . . 2 63 is an example of a Geometric sequence A sequence is geometric if each term r previous term where r is a constant called the common ratio In the above sequence, r = 2 Geometric Sequences and Series A geometric sequence or geometric progression (G.P.) is of the form a, ar , ar 2 , ar 3 , . . . The nth term of an G.P. is un ar n 1 Geometric Sequences and Series Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences (a) 2, 8, 32, . . . Ans: (b) 12, 3, 3 , . . . 4 6th term 2(4) 5 2048 5th term 4 3 1 Ans: 12 64 4 (c) 0.2, 0 02, 0 002, . . . 7th term Ans: 0 2(0.1) 6 0.0000002 Geometric Sequences and Series Summing terms of a G.P. e.g.1 Evaluate 5 3( 2) n 1 n Writing out the terms helps us to recognize the G.P. 3(2) 3(2) 2 3(2) 3 3(2) 4 3(2) 5 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. Geometric Sequences and Series Summing terms of a G.P. With 5 terms of the general G.P., we have TRICK Multiply by r: S 5 a ar ar 2 ar 3 ar 4 rS 5 ar ar 2 ar 3 ar 4 ar 5 Subtracting the expressions gives S 5 rS 5 a ar ar 2 ar 3 ar 4 ar ar 2 ar 3 ar 4 ar 5 Move the lower row 1 place to the right Geometric Sequences and Series Summing terms of a G.P. With 5 terms of the general G.P., we have S 5 a ar ar ar ar 2 Multiply by r: 3 4 rS 5 ar ar ar ar ar 2 3 4 5 Subtracting the expressions gives S 5 rS 5 a ar ar ar ar 2 3 4 5 ar ar ar ar ar 2 and subtract 3 4 Geometric Sequences and Series Summing terms of a G.P. With 5 terms of the general G.P., we have S 5 a ar ar ar ar 2 Multiply by r: 3 4 rS 5 ar ar ar ar ar 2 3 4 5 Subtracting the expressions gives S 5 rS 5 a ar ar ar ar 2 3 4 5 ar ar ar ar ar 2 S 5 rS 5 a 3 4 ar 5 Geometric Sequences and Series Summing terms of a G.P. So, S 5 rS 5 a ar 5 Take out the common factors S5 ( 1 r ) a ( 1 r 5 ) and divide by ( 1 – r ) a( 1 r ) S5 1 r 5 Similarly, for n terms we get a( 1 r ) Sn 1 r n Geometric Sequences and Series Summing terms of a G.P. The formula a( 1 rn ) Sn 1 r gives a negative denominator if r > 1 Instead, we can use a ( r n 1 ) Sn r 1 Geometric Sequences and Series Summing terms of a G.P. For our series 3(2) 3(2) 3(2) 3(2) 3(2) 2 a 6, r 2 and n 5 a ( r n 1 ) Using Sn r 1 6( 2 1) Sn 21 5 6 ( 31 ) 1 186 3 4 5 Geometric Sequences and Series Summing terms of a G.P. EX Find the sum of the first 20 terms of the geometric series, 2 6 18 54 . . leaving your answer in index form 3 6 Solution: a 2, r 3 12 n 20 a ( 1 r ) Sn 1 r S 20 2 1 3 1 3 . We’ll simplify this answer without using a calculator Geometric Sequences and Series Summing terms of a G.P. S 20 2 1 3 20 1 3 1 There are 20 minus signs here and 1 more outside the bracket! 2 1 3 20 42 1 3 20 2 Geometric Sequences and Series 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 ) ar ar 4(a ar ) 2 3 Divide by a since the 1st term, a, cannot be zero: r 2 r 3 4(1 r ) 3 2 r r 4r 4 0 Geometric Sequences and Series Summing terms of a G.P. We need to solve the cubic equation 3 2 r r 4r 4 0 Should use the factor theorem: We will do this soon !! f (1) 1 1 4 4 0 (r 1) is not a factor f (1) 1 1 4 4 0 (r 1) is a factor f ( 2) 8 4 8 4 0 ( r 2) is a factor f ( 2) 8 4 8 4 0 ( r 2) is a factor ( r 1)( r 2)( r 2) are the factors Geometric Sequences and Series Summing terms of a G.P. The solution to this cubic equation is therefore 3 2 r r 4r 4 0 (r 1)( r 2 )(r 2 ) 0 Since we were told r 1 we get r 2 Geometric Sequences and Series SUMMARY A geometric sequence or geometric progression (G.P.) is of the form a, ar , ar 2 , ar 3 , . . . The nth term of an G.P. is un ar n 1 The sum of n terms is a (1 r Sn 1 r n ) or a ( r 1 ) Sn r 1 n Geometric Sequences and Series Sum to Infinity IF |r|<1 then a ( 1 ( 1) ) S 1 (r ) n a S 1 r 0 Because (<1)∞ = 0 Geometric Sequences and Series 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. Geometric Sequences and Series Exercises 1. Solution: 2 + 8 + 32 + . . . a 2, r 4, n 15 S 15 a ( r n 1 ) Sn r 1 2 ( 4 15 1 ) 2(4 15 1 ) S 15 41 3 2. Solution: 4 2+1+. . . a 4, r 0 5, n 15 S 15 4 1 0 5 15 1 0 5 a (1 r n ) Sn 1 r S15 2 67 ( 3 s.f. )