Sum of Arithmetic Series

Question

What is the sum of the first sixteen terms of the arithmetic sequence 

1, 5, 9, 13, ... 61?

This is pretty straightforward:

16(1+61)

------------= 496 Easy.....

2

......but what if I was given:

What is the sum of the first sixteen terms of the arithmetic sequence 

1, 5, 9, 13,

..and not given the last term. How can I easily find that information(61), based on the formula I used.

Answer 1

The pattern for each element is:

a(n) = a(n-1) + 4, a(1) = 1

In closed form:

a(n) = 1 + (n-1)4 = 4n-3

Sum of a(n):

S(n) = ∑(4n-3) = 4∑n -3∑i = 4(n)(n+1)/2 - 3n

S(1) = 4-3 = 1

S(2) = 4(2)(3)/2 - 6 = 12-6 = 6 = 1+5

S(3) = 4(3)(4)/2 - 9 = 24 - 9 = 15 = 1+5 + 9

:

s(16) = 4(16)(17)/2--48 = 496

Answer 2

The common difference is D=4 and the first term is A1=1.

The linear function is f(n) = a1 + d*(n-1)

= 1+ 4(n-1) = 1+ 4n-4 = 4n-3

The sum of the first N terms has formula

(A1+An)/2*N

A16 = f(16)= 4*16-3=64-3=61

The sum is (61+1)/2*16= 62/2*16=496