Sum to Infinity in WAEC Maths: The Formula, the Condition and the Traps

"Sum to infinity" questions are among the quickest marks in WAEC Further Mathematics and Mathematics, once you know when the formula is allowed and how to find the common ratio without slipping. This guide covers the formula, the one condition you must check, and the three question types WAEC keeps setting.

The formula, and the condition that comes with it

For a geometric progression (GP) with first term a and common ratio r:

• the n-th term is Tₙ = arⁿ⁻¹
• the sum of the first n terms is Sₙ = a(1 − rⁿ) ÷ (1 − r)
• if −1 < r < 1, the sum to infinity is S∞ = a ÷ (1 − r)

Why the condition? When the size of r is less than 1, every term is smaller than the one before, rⁿ shrinks towards zero, and the sum settles on a fixed value. When r is 1 or more in size, the terms don't shrink, and the series has no sum to infinity. WAEC sometimes tests exactly this.

Step 1 every time: find r and check it

Divide a term by the term before it, twice:

• r = T₂ ÷ T₁ and r = T₃ ÷ T₂

If the two answers differ, the series is not geometric, and the formula doesn't apply.

Type 1: find the sum to infinity

Find the sum to infinity of 18 + 6 + 2 + 2/3 + …

• a = 18
• r = 6 ÷ 18 = 1/3 (check: 2 ÷ 6 = 1/3 ✓)
• 1/3 lies between −1 and 1, so the sum exists.
• S∞ = 18 ÷ (1 − 1/3) = 18 ÷ (2/3) = 27

Type 2: work backwards from the sum

The first term of a GP is 8 and its sum to infinity is 12. Find the common ratio.

• 12 = 8 ÷ (1 − r)
• 1 − r = 8 ÷ 12 = 2/3
• r = 1/3

Always check that your r lies between −1 and 1; if it doesn't, recheck your algebra.

Type 3: recurring decimals as fractions

Express 0.444… as a fraction.

• 0.444… = 0.4 + 0.04 + 0.004 + …
• a = 4/10, r = 1/10
• S∞ = (4/10) ÷ (1 − 1/10) = (4/10) ÷ (9/10) = 4/9

The same method turns 0.272727… into 27/99 = 3/11.

Negative ratios

Find the sum to infinity of 10 − 5 + 2.5 − …

• a = 10, r = −5 ÷ 10 = −1/2. The signs alternate, and that's allowed.
• S∞ = 10 ÷ (1 − (−1/2)) = 10 ÷ (3/2) = 20/3 (about 6.67)

The commonest slip here is writing 1 − 1/2 instead of 1 + 1/2. Put brackets around a negative r before you substitute.

Traps that cost marks

1. Using the formula when it isn't allowed. For 2 + 6 + 18 + … (r = 3) there is no sum to infinity.
2. Taking r from the wrong pair. r is always a later term divided by the earlier term, never the other way round.
3. Mixing up Sₙ and S∞. If the question names a number of terms ("the first 10 terms"), use Sₙ, not S∞.
4. Fraction arithmetic. Dividing by 2/3 means multiplying by 3/2; write it out rather than doing it in your head.

Practise it

Do ten progression questions back to back, checking the ratio twice each time. Practise free on AceWAEC, and use the AI tutor when a step isn't clear.