Question

$3^n{C}_0-8^n{C}_1+{13}^n{C}_2-{18}^n{C}_3+......+n$

• A. $0$
• B. $3^n$
• C. $5^n$
• D. None of these

Answer 1

The correct option is A $0$

We know that

$(1-x{)}^n{=}^n{C}_0{-}^n{C}_{1}x{+}^n{C}_2{x}^2{-}^n{C}_3{x}^3{+}^n{C}_4{x}^4-.......{+}^n{C}_n(-1{)}^n{x}^n$

Put $x=p^5$

So,

$(1-p^5{)}^n{=}^n{C}_0{-}^n{C}_1{p}^5{+}^n{C}_2{p}^{10}{-}^n{C}_3{p}^{15}{+}^n{C}_4{p}^{20}-.......{+}^n{C}_n(-1{)}^n{p}^{5n}$

On multiplying with $p^3$ both sides, we get

$p^3(1-p^5{)}^n{=}^n{C}_0{p}^3{-}^n{C}_1{p}^8{+}^n{C}_2{p}^{13}{-}^n{C}_3{p}^{18}{+}^n{C}_4{p}^{23}-.......{+}^n{C}_n(-1{)}^n{p}^{8n}$

On differentiating w.r.t $p$, we get

$3p^2(1-p^5{)}^n-p^{3}n(1-p^5{)}^{n-1}5p^4=3{}^n{C}_0{p}^2-8{}^n{C}_1{p}^7+13{}^n{C}_2{p}^{12}-18{}^n{C}_3{p}^{17}$

$+23{}^n{C}_4{p}^{22}-.......+8n{}^n{C}_n(-1{)}^n{p}^{8n-1}$

Put $p=1$

$3(1-1{)}^n-n(1-1^5{)}^{n-1}5=3{}^n{C}_0-8{}^n{C}_1+13{}^n{C}_2-18{}^n{C}_3+23{}^n{C}_4-.......+8n{}^n{C}_n(-1{)}^n$

$3{}^n{C}_0-8{}^n{C}_1+13{}^n{C}_2-18{}^n{C}_3+23{}^n{C}_4-.......+8n{}^n{C}_n(-1{)}^n=0$

Hence, this is the answer.