In a geometric progression, each term is found by multiplying the one before by the same number r. The nth term is T_n = ar^(n−1), and the sum of the first n terms is S_n = a(rⁿ − 1) ÷ (r − 1).
This lesson is part of progressions. It follows terms and sums of arithmetic progressions, and the structure is similar.
What do a, r and n stand for?
- a is the first term.
- r is the common ratio. Find it by dividing any term by the one before.
- n is the position of the term you want.
Write these three down first. Then check r by dividing the second term by the first and the third by the second.
Worked example: a term and a sum
Here is an original example. A geometric progression has first term 3 and common ratio 2. Find T₇ and S₇.
T₇. T₇ = 3 × 2⁶ = 3 × 64 = 192.
S₇. S₇ = 3(2⁷ − 1) ÷ (2 − 1) = 3 × 127 ÷ 1 = 381.
Check by adding the terms: 3 + 6 + 12 + 24 + 48 + 96 + 192 = 381. Both agree.
The mistake that costs marks
The common slip is to use the power n instead of n − 1, so T₇ = 3 × 2⁷ = 384. It looks natural because n = 7.
| Step | Wrong | Right |
|---|---|---|
| Power of r | 7 | 6 |
| T₇ | 3 × 128 | 3 × 64 |
| Result | 384 | 192 |
The test is T₁ = a. With the wrong power, T₁ = 3 × 2¹ = 6, which is the second term, not the first. The correct power gives T₁ = 3 × 2⁰ = 3.
Finding r and a from two terms
Suppose T₂ = 12 and T₅ = 324. Find the progression.
- Divide to cancel a: T₅ ÷ T₂ = r³, because three multiplications separate them. So r³ = 324 ÷ 12 = 27.
- Solve: r = 3.
- Substitute: T₂ = ar = 12, so 3a = 12 and a = 4.
- Check: T₅ = 4 × 3⁴ = 4 × 81 = 324. It matches.
For an arithmetic progression you subtract; for a geometric one you divide. The gap between the term numbers (5 − 2 = 3) becomes the power.
A ratio less than 1
Take a = 5 and r = 1/2. Find S₄.
For r < 1, use S₄ = a(1 − r⁴) ÷ (1 − r) = 5 × (1 − 1/16) ÷ (1/2) = 5 × (15/16) × 2 = 75/8 = 9.375.
Check by adding: 5 + 2.5 + 1.25 + 0.625 = 9.375.
Check yourself
A geometric progression has a = 2 and r = −3. Find T₅ and S₄.
Answer
T₅ = 2 × (−3)⁴ = 2 × 81 = 162.
S₄ = 2 × [(−3)⁴ − 1] ÷ (−3 − 1) = 2 × 80 ÷ (−4) = −40.
Check by adding: 2 − 6 + 18 − 54 = −40. Brackets around −3 matter, since −3⁴ would mean −81.
What to study next
If the ratio is a fraction or a small number, the sum can settle down instead of growing. Continue with using the sum to infinity condition, then practise both types in the practice set.
If powers and indices slow you down, revisit indices, surds and logarithms. For a teacher to check your first line on new questions, see online one-to-one Additional Mathematics tuition.