Sequences, Series and Financial Mathematics
Sequences, Series and Financial Mathematics tests patterns, sums, and money growth. Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. Financial questions apply geometric growth to interest, depreciation and savings.
Arithmetic sequences
- nth term: Tₙ = a + (n − 1)d.
- Sum of n terms: Sₙ = n/2[2a + (n − 1)d].
Geometric sequences
- nth term: Tₙ = a r^(n−1).
- Sum: Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1.
- Sum to infinity exists when |r| < 1: S∞ = a/(1 − r).
Finance
- Compound interest: A = P(1 + i)ⁿ, where i is the rate per period as a decimal and n is the number of periods.
- Depreciation: A = P(1 − i)ⁿ.
- For monthly compounding, divide the annual rate by 12 and multiply the number of years by 12.
Worked example
€2,000 is invested at 4% per annum compounded annually for 3 years. A = 2000(1.04)³ = 2000 × 1.124864 = €2,249.73 (to the nearest cent). The interest earned is €249.73.
Traps
- Using 4 instead of 0.04 for the rate.
- Counting the wrong number of periods, especially for monthly payments.
- Applying simple interest when the question says compounded.
- For depreciation, adding the rate instead of subtracting it.