Sequences and Series: Real-World Scenarios (2)

Question

A person deposits ₹$1000$ on the first day of the month. Each subsequent day, the deposit increases by ₹$100$. How much will be deposited in total over $30$ days?

Solution — Step by Step

Final answer: ₹$73{,}500$.

Why This Works

Whenever a quantity grows by a fixed amount each step, AP applies. The sum formula $S_n = \tfrac{n}{2}(\text{first} + \text{last})$ is essentially Gauss’s pairing trick: pair the first and last, second and second-last, etc. Each pair sums to the same value, and there are $n/2$ such pairs.

If the growth were multiplicative (e.g., interest compounding), it’d be a GP — different formula, same logic.

Alternative Method

Use $S_n = \tfrac{n}{2}[2a + (n-1)d] = \tfrac{30}{2}[2000 + 2900] = 15 \times 4900 = 73500$. Same answer; useful when only $a$ and $d$ are given.

Common Mistake