9. The coefficients of the (r – 1)th, rthand (r + 1)thterms in the expansion of (x + 1)n are in the ratio 1 : 3 : 5. Find n and r.

9. The general term of the expansion (x +1)n isTr+1 = nCrxn–r1ri.e. co-efficient of (r+1)th term = nCrSo, co-efficient of (r−1)th term =nC(r–1) – 1 = nCr – 2Similarly, co-efficient of rth term = nCr – 1 Given that, nCr – 2 :nCr – 1 : nCr = 1 : 3 : 5We have, = 13=> n!(r−2)!(n−r+2)! × (r−1)!(n−r+1)!n! = 13=> (r−1)(r−2)!(n−r+1)!(r−2)!(n−r+2)(n−r+1)! = 13=> (r−1)(n−r+2) = 13=> 3r – 3 = n – r + 2=> 3r + r = n + 2 + 3=> 4r = n + 5 -------------- (1)And, = 35=> n!(r−1)!(n−r+1)! × r!(n−r)!n! = 35=> r(r−1)!(n−r)!(r−1)!(n−r+1)(n−r)! = 35=> rn−r+1 = 35=> 5r = 3n – 3r + 3=> 5r + 3r = 3n + 3=> 8r = 3n + 3 ----------------------- (2)Multiplying equation (1) by 2 and subtracting from equation (2) we get,8r – 8r = 3n + 3 – (2n + 10)=> 0 = 3n + 3 – 2n – 10=> 0 = n – 7=>n = 7Putting n = 7 in equation (1) we get,=> 4r = 7 + 5=>r = 124=>r = 3

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9. The coefficients of the (r – 1)^th, r^thand (r + 1)^thterms in the expansion of (x + 1)ⁿ are in the ratio 1 : 3 : 5. Find n and r.

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Payal Gupta | Contributor-Level 10

4 months ago

9. The general term of the expansion (x +1)ⁿ is

T_r₊₁ = ⁿC_rxⁿ^–r1^r

i.e. co-efficient of ${(r + 1)}^{t h}$ term = ⁿC_r

So, co-efficient of ${(r - 1)}^{t h}$ term =ⁿC_{(r–1) – 1} = ⁿC_r_{– 2}

Similarly, co-efficient of rth term = ⁿC_r_{– 1}

Given that, ⁿC_r_{– 2}:ⁿC_r_{– 1}: ⁿC_r = 1 : 3 : 5

We have,

= $\frac{1}{3}$

=> $\frac{n!}{(r - 2)! (n - r + 2)!}$ × $\frac{(r - 1)! (n - r + 1)!}{n!}$ = $\frac{1}{3}$

=> $\frac{(r - 1) (r - 2)! (n - r + 1)!}{(r - 2)! (n - r + 2) (n - r + 1)!}$ = $\frac{1}{3}$

=> $\frac{(r - 1)}{(n - r + 2)}$ = $\frac{1}{3}$

=> 3r – 3 = n – r + 2

=> 3r + r = n + 2 + 3

=> 4r = n + 5 -------------- (1)

And,

= $\frac{3}{5}$

=> $\frac{n!}{(r - 1)! (n - r + 1)!}$ × $\frac{r! (n - r)!}{n!}$ = $\frac{3}{5}$

=> $\frac{r (r - 1)! (n - r)!}{(r - 1)! (n - r + 1) (n - r)!}$ = $\frac{3}{5}$

=> $\frac{r}{n - r + 1}$ = $\frac{3}{5}$

=> 5r = 3n – 3r + 3

=> 5r + 3r = 3n + 3

=> 8r = 3n +

...more

9. The general term of the expansion (x +1)ⁿ is

T_r₊₁ = ⁿC_rxⁿ^–r1^r

i.e. co-efficient of ${(r + 1)}^{t h}$ term = ⁿC_r

So, co-efficient of ${(r - 1)}^{t h}$ term =ⁿC_{(r–1) – 1} = ⁿC_r_{– 2}

Similarly, co-efficient of rth term = ⁿC_r_{– 1}

Given that, ⁿC_r_{– 2}:ⁿC_r_{– 1}: ⁿC_r = 1 : 3 : 5

We have,

= $\frac{1}{3}$

=> $\frac{n!}{(r - 2)! (n - r + 2)!}$ × $\frac{(r - 1)! (n - r + 1)!}{n!}$ = $\frac{1}{3}$

=> $\frac{(r - 1) (r - 2)! (n - r + 1)!}{(r - 2)! (n - r + 2) (n - r + 1)!}$ = $\frac{1}{3}$

=> $\frac{(r - 1)}{(n - r + 2)}$ = $\frac{1}{3}$

=> 3r – 3 = n – r + 2

=> 3r + r = n + 2 + 3

=> 4r = n + 5 -------------- (1)

And,

= $\frac{3}{5}$

=> $\frac{n!}{(r - 1)! (n - r + 1)!}$ × $\frac{r! (n - r)!}{n!}$ = $\frac{3}{5}$

=> $\frac{r (r - 1)! (n - r)!}{(r - 1)! (n - r + 1) (n - r)!}$ = $\frac{3}{5}$

=> $\frac{r}{n - r + 1}$ = $\frac{3}{5}$

=> 5r = 3n – 3r + 3

=> 5r + 3r = 3n + 3

=> 8r = 3n + 3 ----------------------- (2)

Multiplying equation (1) by 2 and subtracting from equation (2) we get,

8r – 8r = 3n + 3 – (2n + 10)

=> 0 = 3n + 3 – 2n – 10

=> 0 = n – 7

=>n = 7

Putting n = 7 in equation (1) we get,

=> 4r = 7 + 5

=>r = $\frac{12}{4}$

=>r = 3

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Similar Questions for you

If ²⁰C_r is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $\sum_{r = 0}^{20} r^{2}$ ²⁰C_r is equal to

Vishal Baghel

Kindly consider the following figure

If the coefficient of x¹⁰ in the binomial expansion of ${(\frac{\sqrt[]{x}}{5^{\frac{1}{4}}} + \frac{\sqrt[]{5}}{x^{\frac{1}{3}}})}^{60}$ is $5^{k} . l,$ where $l, k \in N a n d l$ is co-prime to 5, then k is equal to………...

alok kumar singh

$T_{r + 1} =^{60} C_{r} {(x^{\frac{1}{2}})}^{60 - r} {(x^{- \frac{1}{3}})}^{r} {(5^{\frac{- 1}{4}})}^{60 - r} {(5^{\frac{1}{2}})}^{r}$

for

$x^{10} \frac{60 - r}{2} - \frac{r}{3} = 10$

$\Rightarrow 180 - 3 r - 2 r = 60$

->r = 24

k = 3 + exponent of 5 in

= $3 + ([\frac{60}{5}] + [\frac{60}{5^{2}}] - [\frac{24}{5}] - [\frac{24}{5^{2}}] - [\frac{36}{5}] - [\frac{36}{5^{2}}])$

= 3 + (12 + 2 – 4 – 0 – 7 – 1)

= 3 + 2 = 5

15. Find the expansion of (3x²– 2ax + 3a²)³using binomial theorem.

Payal Gupta

15. ${[3 x^{2} - 2 a x + 3 a^{2}]}^{3}$

= ${[3 x^{2} + a (- 2 x + 3 a)]}^{3}$

We know that by binomial theorem,

${(a + b)}^{3}$ = $a^{3} + b^{3} + 3 a b (a + b)$

= $a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$

Then,

${[3 x^{2} + a (- 2 x + 3 a)]}^{3}$

= (3x²)³ + ${[a (- 2 x + 3 a)]}^{3}$ + $[3 (3 x^{2})^{2} a (- 2 x + 3 a)]$ + $[3 (3 x^{2}) {a (- 2 x + 3 a)}^{2}]$

= 27x⁶ + $[a^{3} {(- 2 x + 3 a)}^{3}]$ + $[3 (9 x^{4}) (- 2 a x + 3 a^{2})]$ + $[3 (3 x^{2}) {a^{2} {(3 a - 2 x)}^{2}}]$

= 27x⁶ + $[a^{3} {- 8 x^{3} + 27 a^{3} + 3 (4 x^{2}) (3 a) + 3 (- 2 x) (9 a^{2})}]$ + $[- 54 a x^{5} + 81 a^{2} x^{4}]$ + [ $(9 a^{2} x^{2})$ $(9 a^{2} + 4 x^{2} - 12 a x)$ ]

= 27x⁶ + [ $- 8 a^{3} x^{3} + 27 a^{6} + 36 a^{4} x^{2} - 54 a^{5} x$ ] + [ $- 54 a x^{5} + 81 a^{2} x^{4}$ ] + [ $(81 a^{4} x^{2} + 36 a^{2} x^{4} - 108 a^{3} x^{3}$ ]

= 27x⁶ $- 8 a^{3} x^{3} + 27 a^{6} + 36 a^{4} x^{2} - 54 a^{5} x$ $- 54 a x^{5} + 81 a^{2} x^{4}$ + $81 a^{4} x^{2} + 36 a^{2} x^{4} - 108 a^{3} x^{3}$

= 27x⁶– 54ax⁵ $+ 117 a^{2} x^{4}$ $- 116 a^{3} x^{3}$ + $117 a^{4} x^{2}$ $- 54 a^{5} x$ $+ 27 a^{6}$

15. Find the expansion of (3x²– 2ax + 3a²)³using binomial theorem.

Payal Gupta

15. ${[3 x^{2} - 2 a x + 3 a^{2}]}^{3}$

= ${[3 x^{2} + a (- 2 x + 3 a)]}^{3}$

We know that by binomial theorem,

${(a + b)}^{3}$ = $a^{3} + b^{3} + 3 a b (a + b)$

= $a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$

Then,

${[3 x^{2} + a (- 2 x + 3 a)]}^{3}$

= (3x²)³ + ${[a (- 2 x + 3 a)]}^{3}$ + $[3 (3 x^{2})^{2} a (- 2 x + 3 a)]$ + $[3 (3 x^{2}) {a (- 2 x + 3 a)}^{2}]$

= 27x⁶ + $[a^{3} {(- 2 x + 3 a)}^{3}]$ + $[3 (9 x^{4}) (- 2 a x + 3 a^{2})]$ + $[3 (3 x^{2}) {a^{2} {(3 a - 2 x)}^{2}}]$

= 27x⁶ + $[a^{3} {- 8 x^{3} + 27 a^{3} + 3 (4 x^{2}) (3 a) + 3 (- 2 x) (9 a^{2})}]$ + $[- 54 a x^{5} + 81 a^{2} x^{4}]$ + [ $(9 a^{2} x^{2})$ $(9 a^{2} + 4 x^{2} - 12 a x)$ ]

= 27x⁶ + [ $- 8 a^{3} x^{3} + 27 a^{6} + 36 a^{4} x^{2} - 54 a^{5} x$ ] + [ $- 54 a x^{5} + 81 a^{2} x^{4}$ ] + [ $(81 a^{4} x^{2} + 36 a^{2} x^{4} - 108 a^{3} x^{3}$ ]

= 27x⁶ $- 8 a^{3} x^{3} + 27 a^{6} + 36 a^{4} x^{2} - 54 a^{5} x$ $- 54 a x^{5} + 81 a^{2} x^{4}$ + $81 a^{4} x^{2} + 36 a^{2} x^{4} - 108 a^{3} x^{3}$

= 27x⁶– 54ax⁵ $+ 117 a^{2} x^{4}$ $- 116 a^{3} x^{3}$ + $117 a^{4} x^{2}$ $- 54 a^{5} x$ $+ 27 a^{6}$

14. If a and b are distinct integers, prove that a – b is a factor of aⁿ– bⁿ, whenever n is a positive integer.

[Hint: write aⁿ= (a – b + b)ⁿ and expand]

Payal Gupta

14. For (a – b) to be a factor of aⁿ – b ⁿwe need to show (aⁿ – bⁿ) = (a – b)k as k is a natural number.

We have, for positive n

aⁿ = ${(a - b + b)}^{n}$ = ${[(a - b) + b]}^{n}$

=>aⁿ = ⁿC₀(a – b)ⁿ + ⁿC₁(a – b)^{n -}¹b + ⁿC₂(a – b)^{n –}²b² + ………… +ⁿC_n_-1 $(a - b) b^{n - 1}$ + ⁿC_nbⁿ

=>aⁿ= ${(a - b)}^{n}$ + ⁿC₁ ${(a - b)}^{n - 1} b$ + ⁿC₂ ${(a - b)}^{n - 2} b^{2}$ + …………….…+ ⁿC_n_-1 $(a - b) b^{n - 1}$ + $b^{n}$ [Since, ⁿC₀ = 1 and ⁿC_n = 1]

=> $a^{n} - b^{n}$ = ${(a - b)}^{n}$ +ⁿC₁ ${(a - b)}^{n - 1} b$ + ⁿC₂ ${(a - b)}^{n - 2} b^{2}$ + ……………… + ⁿC_n_-1 $(a - b) b^{n - 1}$

=> $a^{n} - b^{n}$ = $(a - b)$ [ ${(a - b)}^{n - 1}$ + ⁿC₁ ${(a - b)}^{n - 2} b$ + ⁿC₂ ${(a - b)}^{n - 3} b^{2}$ +………..…… + ⁿC_n_-1 $b^{n - 1}$ ]

=> $a^{n} - b^{n}$ &n

...more

14. For (a – b) to be a factor of aⁿ – b ⁿwe need to show (aⁿ – bⁿ) = (a – b)k as k is a natural number.

We have, for positive n

aⁿ = ${(a - b + b)}^{n}$ = ${[(a - b) + b]}^{n}$

=>aⁿ = ⁿC₀(a – b)ⁿ + ⁿC₁(a – b)^{n -}¹b + ⁿC₂(a – b)^{n –}²b² + ………… +ⁿC_n_-1 $(a - b) b^{n - 1}$ + ⁿC_nbⁿ

=>aⁿ= ${(a - b)}^{n}$ + ⁿC₁ ${(a - b)}^{n - 1} b$ + ⁿC₂ ${(a - b)}^{n - 2} b^{2}$ + …………….…+ ⁿC_n_-1 $(a - b) b^{n - 1}$ + $b^{n}$ [Since, ⁿC₀ = 1 and ⁿC_n = 1]

=> $a^{n} - b^{n}$ = ${(a - b)}^{n}$ +ⁿC₁ ${(a - b)}^{n - 1} b$ + ⁿC₂ ${(a - b)}^{n - 2} b^{2}$ + ……………… + ⁿC_n_-1 $(a - b) b^{n - 1}$

=> $a^{n} - b^{n}$ = $(a - b)$ [ ${(a - b)}^{n - 1}$ + ⁿC₁ ${(a - b)}^{n - 2} b$ + ⁿC₂ ${(a - b)}^{n - 3} b^{2}$ +………..…… + ⁿC_n_-1 $b^{n - 1}$ ]

=> $a^{n} - b^{n}$ = $(a - b)$ k where k = [ ${(a - b)}^{n - 1}$ + ⁿC₁ ${(a - b)}^{n - 2} b$ + ⁿC₂ ${(a - b)}^{n - 3} b^{2}$ +………..…… + ⁿC_n_-1 $b^{n - 1}$ ] is a natural number.

Therefore (a – b) is a factor of aⁿ – bⁿwhere n is positive integer.

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9. The coefficients of the (r – 1)^th, r^thand (r + 1)^thterms in the expansion of (x + 1)ⁿ are in the ratio 1 : 3 : 5. Find n and r.

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