Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) $$a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5$$
A) $$a _ { n } = 29 - 5 n ; 19,14,9,4 , - 6$$
B) $a _ { n } = 19 ; 19,19,19,19,19$
C) $$a _ { n } = 19 ( n - 1 ) ; 14,9,4 , - 1 , - 6$$
D) $a _ { n } = 19 - 5 n ; 19,14,9,4 , - 1$
E) $a _ { n } = 24 - 5 n ; 19,14,9,4 , - 1$

Question

Quizplus · Accepted Answer

The first five terms of the sequence are:
1, 3, 6, 10, 15

The pattern is that each term is the sum of the previous term and the term number. 
So, the nth term can be written as: 
a(n) = a(n-1) + n

Plugging in the initial value, a(1) = 1:
a(n) = a(n-1) + n = a(n-2) + (n-1) + n = a(n-3) + (n-2) + (n-1) + n = ... = a(1) + 2 + 3 + ... + (n-1) + n

Using the formula for the sum of an arithmetic sequence, 1 + 2 + 3 + ... + (n-1) + n = n(n+1)/2, we can simplify this to:
a(n) = 1 + n(n+1)/2

Therefore, the nth term can be written as a function of n: 
a(n) = 1 + n(n+1)/2, which matches choice E.

Write the First Five Terms of the Sequence Defined Recursively a1=19,ak+1=ak−5a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5a1​=19,ak+1​=ak​−5

Write the First Five Terms of the Sequence Defined Recursively $a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5$