Sequence and Series in Finance
Sequence and series questions test whether you can recognise a pattern, write its nth term and sum it, and then apply that to financial growth. Arithmetic progressions model fixed increments; geometric progressions model percentage growth, compound interest and annuities.
Core rules
- AP: nth term aₙ = a + (n − 1)d; sum Sₙ = n/2[2a + (n − 1)d] = n/2(a + l).
- GP: nth term aₙ = ar^(n−1); sum Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; infinite sum S∞ = a/(1 − r) only when |r| < 1.
- A compound growth multiplier is 1 + i, where i is the rate per period as a decimal. A sum invested for n periods follows a GP with ratio 1 + i.
- An ordinary annuity’s future value is the sum of a GP of payments, so recognising the ratio and number of terms is often faster than listing every payment.
Worked example
A company pays a starting annual salary of ₹18,000 and increases it by ₹1,200 each year. This is an AP with a = 18,000 and d = 1,200. The fifth-year salary is a₅ = 18,000 + (5 − 1)(1,200) = ₹22,800. Total salary over five years is S₅ = 5/2[2(18,000) + 4(1,200)] = 2.5(36,000 + 4,800) = 2.5 × 40,800 = ₹102,000. The common error is to use n = 5 in the nth-term formula without subtracting 1; the fifth year contains only four increments.
Traps
- Using a + nd instead of a + (n − 1)d.
- Applying the infinite GP sum when |r| ≥ 1.
- Treating a percentage growth rate as the ratio itself instead of 1 + r/100.
- Mixing the number of payments with the number of periods when payments occur at the beginning of each period.