Binomial Theorem

Maths · Class 11

Simulation · Maths · Class 11

Estimate (1 + x)¹⁰ term by term

"Estimate (1 + x)¹⁰ term by term" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Numbers, estimates and remainders" in Binomial Theorem. (1 + x)¹⁰ = 1 + 10x + 45x² + 120x³ + … Slide x and the number of terms kept. Each coloured piece is one term; the dashed line is the exact value. For small x, (1 + x)ⁿ ≈ 1 + nx + nC2 x². The smaller x is, the faster the terms shrink and the fewer you need. JEE Main asked 54 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Numbers, estimates and remainders in Binomial Theorem. Change the values and watch what happens.

Estimate (1 + x)¹⁰ term by termMaths · Class 11

In JEE Main

42 shifts analysed

54 questions from Binomial Theorem across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §7.2.2

  • Large powers: write the base as a round number plus or minus a small one. 98⁵ = (100 − 2)⁵ = 10000000000 − 1000000000 + 40000000 − 800000 + 8000 − 32 = 9039207968.
  • Our own check: 102⁴ = (100 + 2)⁴ = 100000000 + 8000000 + 240000 + 3200 + 16 = 108243216.
  • Comparing sizes: (1.01)¹⁰⁰⁰⁰⁰⁰ = (1 + 0.01)¹⁰⁰⁰⁰⁰⁰ = 1 + 1000000 × 0.01 + (other positive terms) = 10001 + (positive terms), so it is larger than 10000.
  • Estimates: for small x, the first terms of (1 + x)ⁿ give a close value. (1.02)¹⁰ ≈ 1 + 10(0.02) + 45(0.02)² = 1 + 0.2 + 0.018 = 1.218, against the true value 1.21899…; each further term is much smaller than the one before.
  • Remainders: 6ⁿ − 5n leaves remainder 1 on division by 25. Expanding (1 + 5)ⁿ = 1 + 5n + 5²(nC2) + 5³(nC3) + …, so 6ⁿ − 5n = 1 + 25 × (a whole number).
  • The same method shows 7ⁿ − 6n − 1 is divisible by 36: (1 + 6)ⁿ = 1 + 6n + 36(nC2 + 6·nC3 + …).
  • The general idea: to find a power's remainder on division by m, write the base as (multiple of m) ± 1 or ± a small number, and expand.

Questions

3 answers

What does the "Estimate (1 + x)¹⁰ term by term" simulation show?
(1 + x)¹⁰ = 1 + 10x + 45x² + 120x³ + … Slide x and the number of terms kept. Each coloured piece is one term; the dashed line is the exact value.
Does JEE Main ask questions from Binomial Theorem?
Yes. JEE Main asked 54 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Numbers, estimates and remainders" in Binomial Theorem (Class 11 Maths).

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