Geometric series
- 2. To find the sum of the first n terms of a geometric series, we use the following formulas: Sn = A1 (1 – rn ) 1 – r , r ≠ 1 and Sn = A1 − rXn 1 – r , r ≠ 1 Where A1 = first term An = nth term r = common ratio Sn = sum of the first n terms
- 3. Example 1. Find the sum of the first 12 terms of the geometric sequence 4, 16, 64, …... Steps Solution 1. Enumerate the given information. Here A1 = 4 2. Find the common ratio using the formula r = An An − 1 r = 16 4 = 4
- 4. Steps Solution 3. Solve for the sum of the first 12 terms using the formula Sn = A1 (1 – rn ) 1 – r , r ≠ 1 Sn = A1 (1 – rn ) 1 – r , r ≠ 1 S12 = 4 (1 – 412 ) 1 − 4 , r ≠ 1 = 4(1 – 16,777,216) −3 = 4( –16,777,215 ) −3 = 4(5, 592, 405) S12 = 22,369,620
- 5. Example 2. Find the sum of the first seven terms of the geometric sequence -1, -5, -25, -125,….. Steps Solution 1. Enumerate the given information. Here A1 = -1 2. Find the common ratio. r = −5 −1 = 5
- 6. Steps Solution 3. Solve for the sum of the first seven terms using the formula Sn = A1 (1 – rn ) 1 – r , r ≠ 1 Sn = A1 (1 – rn ) 1 – r , r ≠ 1 S7 = −1 (1 – 57 ) 1 – 5 , S7 = −1 ( 1 – 78,125 ) −4 S7 = −1 (–78,124) −4 = 𝟕𝟖,𝟏𝟐𝟒 −𝟒 S7 = -19,531
- 7. Example 3. Find the sum of the first ten terms of the geometric sequence whose first term is 1 2 and whose fourth term is 4. Steps Solution 1. Enumerate the given information. Here, A1 = 1 2 and A4 = 4 2. Find the common ratio. If A1 = 1 2 and A4 = 4, we have An = A1rn-1 A4 = A1r4-1 4 = 1 2 r3 4 1 2 = 4 . 2 1 = 8 8 = r3 ; 3 8 = r 2 = r
- 8. Steps Solution 3. Solve for the sum of the first ten terms using the formula Sn = A1 (1 – rn ) 1 – r , r ≠ 1 If A1 = 1 2 , A4 = 4, r = 2, n = 10, we have S10 = 1 2 (1 – 210 ) 1 – 2 S10 = 1 2 (1 – 1,024) −1 S10 = 1(−1,023) 2 ÷ -1 S10 = −1,023 2 ÷ -1 S10 = 𝟏,𝟎𝟐𝟑 𝟐
- 9. Activity
- 10. Find the needed term of the given geometric sequence 1. 5, 10, 20, 40, …, find S10 Steps Solution 1. Enumerate the given information. A1 = 5 ; n = 10 ; r = 10 5 = 2 2. Solve for the sum of the first ten terms using the formula Sn = A1 (1 – rn ) 1 – r , r ≠ 1 S10 = 5(1 – 210 ) 1 – 2 S10 = 5(1 – 1,024 ) −1 S10 = 5(−1,023 ) −1 = −5,115 −1 S10 = 5,115
- 11. 2. 960, 240, 60, 15, …, find S5 Steps Solution 1. Enumerate the given information. A1 = 960 ; n = 5 ; r = 240 960 = 1 4 2. Solve for the sum of the first five terms using the formula Sn = A1 (1 – rn ) 1 – r , r ≠ 1 S5 = 960(1 – ( 1 4 )5 1 – 1 4 S5 = 960(1 – 1 1,024 ) 3 4 S5 = 𝟓,𝟏𝟏𝟓 𝟒