math precalculus 12

Question

If an arithmetic series has a = 13.7 and t₃ = 12.3, then determine S₃.

Answer 1

Dylan, these problems are really simple. Know that a = the first term and t₃ = the third term of an arithmetic sequence.

In an Arithmetic sequence, the difference between the terms is the same. If the first term is 3, and the second term is 7, then the difference between all terms is 4. The sequence would be 3 + 7 + 11 + 15 + ...

Since we have the first and third terms and want to find S₃, we can use the formula S_n = n/2(a₁ + a_n)

S3 = (3/2)(13.7 + 12.3) = 39.0

Without the formula we can do it brute force method.

In your problem, the first term is 13.7. The third term is 12.3. We can find the 2nd term which is halfway between the first and the third. (13.7 - 12.3)/2 = 1.4/2 = 0.7 = the difference between terms.

So the first term is 13.7, the second term 13.7 - 0.7 = 13, and the third term, 13 - 0.7 = 12.3.

S3 is the sum of the first 3 terms, 13.7 + 13 + 12.3 = 39.0.

If we were looking for the 150th term this wouldn't be feasible. We'd need to use the formula.

Answer 2