Q.Find the future value of an ordinary annuity of ₹1,000 per year for 4 years at 10% p.a. compounded annually, using the GP sum formula.

Each ₹1,000 instalment grows with interest for a different number of remaining years, so the accumulated values of all instalments together form a GP whose sum gives the annuity's future value.

Given: ordinary annuity, $R = ₹1{,}000$ per year, $i = 10\% = 0.10$, $n = 4$ years.

Step 1 — Recognise the GP: the accumulated value of each instalment (the last one earning no interest, the first one earning interest for 3 years) forms a GP with $a = R = 1000$ and $r = (1+i) = 1.10$, summed over $n=4$ terms.

Step 2 — Apply the annuity future-value formula: $FV = R \cdot \dfrac{(1+i)^n - 1}{i} = 1000 \times \dfrac{(1.10)^4 - 1}{0.10}$.

Step 3 — Compute $(1.10)^4$: $(1.10)^2 = 1.21$; $(1.10)^4 = 1.21^2 = 1.4641$.

Step 4 — Substitute and simplify: $\dfrac{1.4641 - 1}{0.10} = \dfrac{0.4641}{0.10} = 4.641$.

Step 5 — Multiply by $R$: $FV = 1000 \times 4.641 = 4641$. …

Applying the compound-interest single-sum formula $A=P(1+i)^n$ instead of the annuity sum formula — an annuity involves a SERIES of payments, so it needs …