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Using binomial theorem, prove that (2^(3...

Using binomial theorem, prove that `(2^(3n)-7n-1)` is divisible by 49, where n `in` N.

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`( 1 + x ) ^ ( n ) = . ^ ( n) C _ ( 0 ) + .^ (n) C _ ( 1 ) x + .^ ( n ) C _ ( 2 ) x ^ ( 2 ) + ...+ .^ ( n ) C _ ( n)x ^ ( n ) .`
`( 1 + 7 ) ^ ( n ) = . ^ ( n) C _ ( 0 ) + . ^ ( n) C_ ( 1 ) xx 7 + . ^ ( n ) C _( 2 ) xx ( 7 ) ^ ( 2 ) + ...+ . ^ ( n ) C _ ( n ) xx ( 7 ) ^ ( n ) `
`= 1 + 7 n + ( 7 ) ^ ( 2 ) [.^ ( n) C_ ( 2 ) + .^ ( n) C_ ( 3 ) xx 7 + ... + ( 7 ) ^ ( n-2 ) ]`
`rArr ( 8 ^ ( n ) - 7n _ 1 ) = ( 7 ) ^ ( 2 ) xx " (an integar)" = 49 xx "( an integar) ".` ,
Hence, `(2 ^ ( 3n ) -7n-1)` is divisible by 49.
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RS AGGARWAL-BINOMIAL THEOREM-EXERCISE 10A
  1. Prove that sum(n)^(r=0) ""^(n)C(r)*3^(r)=4^(n).

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  2. Using binomial theorem, evaluate each of the following: (i)(104)^(4...

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  3. Using binomial theorem, prove that (2^(3n)-7n-1) is divisible by 49, w...

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  4. Prove that (2+sqrt(x))^(4)+(2-sqrt(x))^(4)= 2(16+24x+x^(2)).

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  5. Find the 7th term in the expansion of ((4x)/5+5/(2x))^(8)

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  6. Find the 9th term in the expansion of (a/b-b/(2a)^(2))^(12).

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  7. Find the 16th term in the expansion of (sqrt(x)-sqrt(y))^(17)

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  8. Find the 13^(t h)term in the expansion of (9x-1/(3sqrt(x)))^(18),x!=0

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  9. If the coefficients of x^7 and x^8 in the expansion of [2 +x/3]^n a...

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  10. The ratio of the coefficient of x^(15) to the term independent of x in...

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  11. Prove that the ratio of the coefficient of x^10 in (1 - x^2)^10 & the ...

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  12. Find the term independent of x in the expansion of (1+x+2x^3)[(3x^2//2...

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  13. Find the coefficient of :\ x\ in the expansion of (1-3x+7x^2)(1-x)^(1...

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  14. Find the coefficient of (i) x^(5)" in the expansion of "(x+3)^(8) (...

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  15. Show that the term containing to does not exist in the expansion of (3...

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  16. Does the expansion of (2x^2-1/x)^(20) contain any term involving x^9?

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  17. Show that the expansion of (x^2+1/x)^1 does not contain any term invol...

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  18. Write the general term in the expansion of (x^(2)-y)^(6).

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  19. Find the 5th term from the end in the expansion of (x-1/x)^(12).

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  20. Find the 4th term from the end in the expansion of ((4x)/5-5/(2x))^9do...

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