How many terms of the series 25+19+13 +… are required to make the sum −20?

Question

How many terms of the series 25+19+13 +… are required to make the sum −20?

Answer 1

If term 1 is a₁, then, for an arithmetic series a_n = (n-1)d + a₁ . The sum of the a's from 1 to n will be na₁ + (1+2+3+...n-1) d. This last expression can be written as na₁ + (n-1)(n)/2 * d = S_n

In this case, S_n = -20, a₁ = 25 and d = -6. This leads to a quadratic in n. (I get n = 10)

Answer 2

Use the arithmetic sum formula:

s_n = (n/2) [2a+ (n-1)d]

n= term number = ???= What we will solve for

a= stands for the 1st term= 25

d= common difference = -6

s_n =-20

Remember that the number is a natural number, because you will have to solve a quadratic equation and will have 2 solutions.