Question

If 2ⁿ⁺² – 2ⁿ⁺¹ + 2ⁿ = c × 2ⁿ, find the value of c.

Answer 1

We have, 2ⁿ⁺² + 2ⁿ⁺¹ + 2n = c × 2ⁿ

⇒ 2ⁿ2² + 2ⁿ2¹ + 2ⁿ = c × 2ⁿ  ...[∴ a^m+n = a^m × aⁿ]

⇒ 2ⁿ[2² – 2¹ + 1] = c × 2ⁿ  ...[Taking common 2ⁿ in LHS]

⇒ 2ⁿ[4 – 2 + 1] = c × 2ⁿ

⇒ 3 × 2ⁿ  = c × 2ⁿ

3 × 2ⁿ × 2^–n = c × 3ⁿ × c^–n  ...[Multiplying both sides by 2^–n]

⇒ 3 × 2^n–1 = c × 2^n–n   ...[∴ a^m+n = a^m × aⁿ]

⇒ 3 × 2⁰ = c × 2⁰

⇒ 3 × 1 = c × 1  ...[∴ a⁰ = 1]

∴ 3 = c