Q.Madhu exchanged her old car valued at ₹ 1,50,000 with a new one priced at ₹ 6,50,000. She paid ₹ x as down payment and the balance in 20 monthly equal instalments of ₹ 21,000 each. The rate of interest offered to her is 9% p.a. Find the value of x. [Given that : $(1 \cdot 0075)^{-20} = 0 \cdot 86118985$]

Annuities: The Intuition First

Imagine you're saving for retirement. You decide to put ₹10,000 into a fixed deposit every year for 20 years. At the end of each year, the bank gives you 8% interest on whatever is in your account. What will your total savings be at the end of 20 years?

That's an annuity — a series of equal payments made at regular intervals. The key idea is that each payment earns interest for a different length of time. The first ₹10,000 earns interest for 20 years, the second for 19 years, and the last one earns interest for just 1 year (or zero, depending on when you count).

Two Kinds of Annuity Value

There are two questions you can ask:

1. Future Value (FV) — "If I deposit ₹P every year for n years at r% interest, how much will I have at the end?"
2. Present Value (PV) — "If someone promises to pay me ₹P every year for n years, and I can earn r% elsewhere, what is that promise worth right now?"

Both are built from the same core idea: each payment is a separate compound-interest problem, and you add them up.

The Precise Mathematics

Let:

• $P$ = the regular payment (the annuity amount)
• $r$ = interest rate per period (as a decimal; e.g., 8% = 0.08)
• $n$ = number of payments

Future Value of an Ordinary Annuity

"Ordinary" means payments happen at the end of each period. The first payment earns interest for $n-1$ periods, the second for $n-2$, and the last earns no interest at all.

This is a geometric series. Factor out $P$ and sum it:

Present Value of an Ordinary Annuity

Here we discount each future payment back to today. The first payment (one period away) is discounted by $(1+r)^1$, the second by $(1+r)^2$, and so on.

Again, a geometric series:

A Concrete Example

You want to know how much ₹10,000 saved every year for 20 years at 8% will grow to.

$P = 10000$, $r = 0.08$, $n = 20$

$(1.08)^{20} \approx 4.66096$

$FV = 10000 \cdot \frac{4.66096 - 1}{0.08} = 10000 \cdot \frac{3.66096}{0.08} = 10000 \cdot 45.762 = 457,620$ …