For integers n and r , let n r = C r n , if n ≥ r ≥ 0 0 , otherwise . The maximum value of k for which the sum ∑ i = 0 k 10 i 15 k - i + ∑ i = 0 k + 1 12 i 13 k + 1 - i is maximum, is equal to _________.

Question

For integers n and r , let n r = C r n , if n &#8805; r &#8805; 0 0 , otherwise . The maximum value of k for which the sum &#8721; i = 0 k 10 i 15 k - i + &#8721; i = 0 k + 1 12 i 13 k + 1 - i is maximum, is equal to _________.

Quantrex Academy · Accepted Answer

Correct answer: 0

( 1 + x ) 10 = C 0 10 + C 1 10 x + C 2 10 x 2 + &#8230; &#8230; + C 10 10 x 10 ( 1 + x ) 15 = C 0 15 + C 1 15 x + &#8230; . C k - 1 15 x k - 1 + C k 15 x k + C k + 1 15 x k + 1 + &#8230; &#8230; 15 C 15 x 15 &#8721; i = 0 k 10 i 15 k - i = C 0 10 . C k 15 + C 1 10 . C k - 1 15 + &#8230; . + C k 10 . C 0 15 Coefficient of x k in ( 1 + x ) 25 = C k 25 &#8721; i = 0 k + 1 12 i 13 k + 1 - i = &#8721; i = 0 k + 1 C i 12 C k + 1 - i 13 = C 0 12 . C k + 1 13 + C 1 12 . C k 13 + &#8230; . . + C k + 1 12 . C 0 13 Coefficien

For integers n and r , let n r = C r n , if n ≥ r ≥ 0 0 , otherwise . The maximum value of k for which the sum ∑ i = 0 k 10 i 15 k - i + ∑ i = 0 k + 1 12 i 13 k + 1 - i is maximum, is equal to _________.

Correct answer

Step-by-step solution

More from Binomial Theorem

For integers n and r , let n r = C r n , if n &#8805; r &#8805; 0 0 , otherwise . The maximum value of k for which the sum &#8721; i = 0 k 10 i 15 k - i + &#8721; i = 0 k + 1 12 i 13 k + 1 - i is maximum, is equal to _________.

Correct answer

Step-by-step solution

More from Binomial Theorem

For integers n and r , let n r = C r n , if n ≥ r ≥ 0 0 , otherwise . The maximum value of k for which the sum ∑ i = 0 k 10 i 15 k - i + ∑ i = 0 k + 1 12 i 13 k + 1 - i is maximum, is equal to _________.