Finding the sum of a geometric sequence
- 1. EXAMPLE 1: Find the sum of the sequence: 100, 25, 5, 1, 1/5,…
- 2. Find the sum of the sequence: 100, 25, 5, 1, 1/5,… First, figure out if the sequence is arithmetic or geometric. It is GEOMETRIC because you are MULTIPLYING by 1/5 each time.
- 3. Find the sum of the sequence: 100, 25, 5, 1, 1/5,… Next, write the explicit formula for this sequence. (This is not needed to solve the problem, it is just good practice!) 𝑎 𝑛 = 𝑎1 ∙ 𝑟 𝑛−1 Since the first term is 100, that is plugged in for a1. The common ratio is 1/5 The new equation is: 𝑎 𝑛 = 100 ∙ ( 1 5 ) 𝑛−1
- 4. Find the sum of the sequence: 100, 25, 5, 1, 1/5,… Now find the sum! 𝑆 = 𝑎1 1−𝑟 where a1 is the first term and r is the common ratio 𝑆 = 100 1−1/5 = 12.5 The sum of this geometric sequence is 12.5
- 5. EXAMPLE 2: Find the sum of the first 10 terms in the sequence: 3, 6, 12, 24, 48, …
- 6. Find the sum of the first 10 terms in the sequence: 3, 6, 12, 24, 48, … First, figure out if the sequence is arithmetic or geometric. It is GEOMETRIC because you are MULTIPLYING by 2 each time.
- 7. Find the sum of the first 10 terms in the sequence: 3, 6, 12, 24, 48, … Next, write the explicit formula for this sequence. (This is not needed to solve the problem, it is just good practice!) 𝑎 𝑛 = 𝑎1 ∙ 𝑟 𝑛−1 Since the first term is 3, that is plugged in for a1. The common ratio is 2 The new equation is: 𝑎 𝑛 = 3 ∙ (2) 𝑛−1
- 8. Find the sum of the first 10 terms in the sequence: 3, 6, 12, 24, 48, … Now find the sum! 𝑆 = 𝑎1(1−𝑟 𝑛) 1−𝑟 where a1 is the first term, r is the common ratio and n is the number of terms 𝑆 = 3(1−210) 1−2 = −3071 The sum of this geometric sequence is -3071