If 2n+2 +2n+1 + 2n = c x 2n, then find c.

1.	If 2n+2 +2n+1 + 2n = c x 2n, then find c.
Answer» We have, 2ⁿ⁺²+2ⁿ⁺¹ + 2n = c x 2ⁿ => 2ⁿ 2²+2ⁿ2¹ + 2ⁿ = c x 2ⁿ [∴ a^m+n = a^m x aⁿ] => 2ⁿ [2² – 2¹ + 1] = c x 2ⁿ [taking common 2ⁿ in LHS] => 2ⁿ[4-2 + 1]=c x 2ⁿ => 3 x 2ⁿ = c x 2ⁿ 3 x 2ⁿx 2^-n =c x 3ⁿx c^-n [multiplying both sides by 2^-n] => 3 x 2^n-1 = c x 2^n-n [∴ a^m+n = a^m x aⁿ] => 3 x 2° =c x 2° => 3x 1 =c x 1 [∴ a°=1] ∴ 3 = c

Answer»

We have, 2ⁿ⁺²+2ⁿ⁺¹ + 2n = c x 2ⁿ

=> 2ⁿ 2²+2ⁿ2¹ + 2ⁿ = c x 2ⁿ [∴ a^m+n = a^m x aⁿ]

=> 2ⁿ [2² – 2¹ + 1] = c x 2ⁿ [taking common 2ⁿ in LHS]

=> 2ⁿ[4-2 + 1]=c x 2ⁿ

=> 3 x 2ⁿ = c x 2ⁿ

3 x 2ⁿx 2^-n =c x 3ⁿx c^-n [multiplying both sides by 2^-n]

=> 3 x 2^n-1 = c x 2^n-n [∴ a^m+n = a^m x aⁿ]

=> 3 x 2° =c x 2°

=> 3x 1 =c x 1 [∴ a°=1]

∴ 3 = c

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