show that 9^(n+1) - 8n -9 is divisible by 64 whenever n is a positive integer

 ncert cbse chapter 8 binomial theorem exercise 8.1

13.show that 9^(n+1) - 8n -9 is divisible by 64 whenever n is a positive integer 

(x+1)^(n+1) = [x^(n+1)] +C[(n+1),1][x^(n)]+...+C[(n+1),(n-1)][x^2] +C[(n+1),n][x] + 1

 

C[(n+1),n] =C[(n+1),1] = (n+1)

C[(n+1),(n-1)]= C[(n+1),2] = (n+1)(n)/2

(x+1)^(n+1) = [x^(n+1)] +(n+1)[x^(n)]+...+[(n+1)(n)/2][x^2]+(n+1)[x] + 1

put x=8

 (8+1)^(n+1) = [8^(n+1)] +(n+1)[8^(n)]+...+[(n+1)(n)/2][8^2]+(n+1)[8] + 1

concentrating on the last 2 terms only

9^(n+1) =  { [8^(n+1)] +(n+1)[8^(n)]+...+[(n+1)(n)/2][8^2]} + [8n+8]+1

9^(n+1) =  { [8^(n+1)] +(n+1)[8^(n)]+...+[(n+1)(n)/2][8^2]} + 8n+9

re-arranging

 9^(n+1) - 8n - 9 ={ [8^(n+1)] +(n+1)[8^(n)]+...+[(n+1)(n)/2][8^2]}

note that C[(n+1),(n-1)]= C[(n+1),2] = (n+1)(n)/2 is an integer

so after taking factor of (8^2) = 64 out,

9^(n+1) - 8n - 9 = 64*[8^(n-1)+ ...+[(n+1)(n)/2] ] =divisible by 64

8. evaluate (101)^4

using expansion with 

C(4,1) =4=C(4,3)

C(4,2)=6

(x+y)^4 = (x^4)+4(x^3)(y)+6(x^2)(y^2)+4(x)(y^3)+(y^4)

x=100

y=1

(100+1)^4 = (100^4)+4(100^3)(1)+6(100^2)(1^2)+4(100^1)(1^3)+(1^4)

(101)^4 =100000000+4000000+60000+400+1

(101)^4 =104060401

 

chapter 8 binomial theorem miscellaneous exercise

 1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

solution

 

2.  Find a if the coefficients of (x^2)  & (x^3) in the expansion of {(3+ax)^9} are equal 

solution

  3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

solution

 

5.evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

solution 

 6.find the value of [(a^2)+sqrt{(a^2)-1}]^4 + [(a^2)-sqrt{(a^2)-1}]^4 

solution

 

7.find an approximate value of (0.99^5) using the first three terms of its expansion

solution  

 

8.find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of [(fourth root of 2) + {1/(fourth root of 3)}]^n is (sqrt6):1 

solution

 

exercise 8.2

 

 5. find the 4th term in the expansion of (x-2y)^12

solution

7. Find the middle terms in the expansion of [3 - ((x^3) / 6)]^7

solution

Q8) Find the  middle terms in the expansion of [(x/3)+9y)]^10

solution

 

 10.The coefficients of the (r-1)th, rth, (r+1)th  terms in the expansion of [(x+1)^n] is in the ratio 1:3:5. Find n and r.

solution 

 

exercise 8.1

8. evaluate (101)^4

solution

13.show that 9^(n+1) - 8n -9 is divisible by 64 whenever n is a positive integer 

solution

 

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