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Doubt & Answer Explanation

Doubt

if (1+x)n = c0 + c1 x + c 2 xx +....cnxn ,then the sum of the series co +5c1 +9c2 +....... +(4n +1)cn will be

Answer Explanation

(1+x)ⁿ = c₀ + c₁ x + c ₂ x² +....c_nxⁿ Let x = y⁴
(1+y4)ⁿ = C₀+ C₁y⁴ + C₂y⁸ + C₃y¹² + ..................................
Multiplying both sides by y
y(1+y⁴)ⁿ = C_oy+C₁y⁵ + C₂y⁹ + C₃y¹³+ ...............
Differentiating both sides , we get
(1+y⁴)ⁿ + y. n(1+y⁴)^n-1 .4y³ = C₀ + 5C₁y⁴ + 9C₂y⁸ + .......................
put y = 1
2ⁿ + n2^n-1.4 = C0 + 5C1 + 9C2 + ................
2ⁿ ( 1+ 2n) = C0 + 5C1 + 9C2 + .........

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Doubt & Answer Explanation

Doubt

if (1+x)*n = c0 + c1 x + c 2 x*x +....cnxn ,then the sum of the series co +5c1 +9c2 +....... +(4n +1)cn will be

Answer Explanation

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if (1+x)n = c0 + c1 x + c 2 xx +....cnxn ,then the sum of the series co +5c1 +9c2 +....... +(4n +1)cn will be