For k ∈ N , let 1 α ( α + 1 ) ( α + 2 ) … … . ( α + 20 ) = ∑ K = 0 20 A k α + k , where α > 0 . Then the value of 100 A 14 + A 15 A 13 2 is equal to ____________.

Question

For k &#8712; N , let 1 &#945; ( &#945; + 1 ) ( &#945; + 2 ) &#8230; &#8230; . ( &#945; + 20 ) = &#8721; K = 0 20 A k &#945; + k , where &#945; &#62; 0 . Then the value of 100 A 14 + A 15 A 13 2 is equal to ____________.

Quantrex Academy · Accepted Answer

Correct answer: 0

We have, &#8721; k = 0 20 A k &#945; + k = 1 &#945; &#945; + 1 &#8230; &#8230; &#945; + 20 &#8658; A 0 &#945; + A 1 &#945; + 1 + . . . + A 20 &#945; + 20 = 1 &#945; &#945; + 1 &#8230; &#8230; &#945; + 20 &#8658; A 0 &#945; + 1 &#945; + 2 &#8230; &#8230; &#945; + 19 &#945; + 20 + A 2 &#945; &#945; + 2 &#8230; &#8230; &#945; + 19 &#945; + 20 + . . . + A 20 &#945; &#945; + 1 &#945; + 2 &#8230; &#8230; &#945; + 19 &#945; &#945; + 1 &#945; + 2 &#8230; &#8230; &#945; + 20 = 1 &#945; &#945; + 1 &#945; + 2 &#8230; &#8230;

For k ∈ N , let 1 α ( α + 1 ) ( α + 2 ) … … . ( α + 20 ) = ∑ K = 0 20 A k α + k , where α > 0 . Then the value of 100 A 14 + A 15 A 13 2 is equal to ____________.

Correct answer

Step-by-step solution

More from Sequences and Series

For k &#8712; N , let 1 &#945; ( &#945; + 1 ) ( &#945; + 2 ) &#8230; &#8230; . ( &#945; + 20 ) = &#8721; K = 0 20 A k &#945; + k , where &#945; &#62; 0 . Then the value of 100 A 14 + A 15 A 13 2 is equal to ____________.

Correct answer

Step-by-step solution

More from Sequences and Series

For k ∈ N , let 1 α ( α + 1 ) ( α + 2 ) … … . ( α + 20 ) = ∑ K = 0 20 A k α + k , where α > 0 . Then the value of 100 A 14 + A 15 A 13 2 is equal to ____________.