Word Problem: The Average of Consecutive Odd Integers

Question

The average of $19$ consecutive odd integers is $539$. Find the smallest of these integers.

A.) $529$

B.) $527$

C.) $521$

D.) $519$

Answer 1

Let x be a number.

$2x+1$ will give you an odd number (Let this be the first odd number of your sequence).

$({(2x+1)+(2x+3)+(2x+5)+...+(2x+37)})/19 = 539$

$38x=9880$

$x=260$

$2x+1=521$

Since $2x+1$ is your first number of the sequence, 521 is the answer (option c).

Answer 2

An arithmetic progression (such as consecutive odd numbers) with an odd number of terms (such as 19) has an average equal to its middle term, since the remaining ones pair up in symmetrical couples that average to the same value.

In your case the average is 539, so the middle term is 539. That's the 10th term out of the total 19. The smallest term would be the 1st, which is 10 - 1 = 9 places back. Since the "common difference" between odd integers is 2, the smallest term then calculates to 539 - 9 * 2 = 521.