Question

24. Four friends attempted to write the explicit expression for the nth term of the sequence 2, 5, 10, 17, ….

24. Four friends attempted to write the explicit expression for the nth term of the sequence 2, 5, 10, 17, ….



If n represents the set of counting numbers, who wrote the correct expression

If n represents the set of counting numbers, who wrote the correct expression?


A. Austin


B. Kaylee


C. Noah


D. Zoey

Answer 1

2+3 = 5
5+5 = 10
10+7 = 17
3, 5, 7 are the first three prime numbers after [1 and] 2. Let p1 = 3, p2=5, p3=7, etc. It is logical to assume that the next terms in the sequence are
17+11 = 28
28+13 = 41
etc.
so that we have
2+p1 = t1 (first term in sequence)
t1+p2 = t2
t2+p3 = t3
...
t(n-1)+pn = tn
or...
2+p1+p2+...+pn = tn
This last is the general term sought.

Answer 2

Or

did you notice that each of the given numbers is one more than the perfect square, that is
2 = 1^2 + 1
5 = 2^2 +1
10 = 3^2 + 1
17 = 4^2 + 1
thus:

term(n) = n^2 + 1