Ramesh saves ₹ 50 in the first week and ₹ 10 more in each week after that. Suresh saves ₹ 100 every week. After how many weeks will their total savings be equal, and how much will each have saved?

1. 1 · Write what is given and what is to be found. Ramesh: a = 50, d = 10. Suresh: ₹ 100 every week. To find: n when the totals are equal, and the total.
2. 2 · Write the formula or theorem. Sₙ = n/2 [2a + (n − 1)d]
3. 3 · Write Ramesh's total. Sₙ = n/2 [100 + 10(n − 1)] = n/2 (10n + 90) = 5n² + 45n
4. 4 · Write Suresh's total. 100 × n = 100n
5. 5 · Set them equal and solve.
   • 5n² + 45n = 100n
   • 5n² − 55n = 0, so 5n(n − 11) = 0
   • n = 0 or n = 11. n = 0 means no weeks have passed, so reject it: n = 11.
6. 6 · Find the total. Suresh: 100 × 11 = ₹ 1100. Ramesh: a₁₁ = 50 + 10 × 10 = 150, so S₁₁ = 11/2 × (50 + 150) = ₹ 1100.
7. 7 · Check and box the answer. After 11 weeks, both have saved ₹ 1100.

Scoring tip: Write each person's total as an expression in n first. Then the equation almost writes itself.