The fifth term of an arithmetic series is 9, and the sum of the first 16 terms is 480. How do you find the first three terms of the sequence?

1 Answer
Jan 24, 2016

Answer:

Solve to find the first term and common difference of the sequence and hence the first three terms:

#15/7#, #27/7#, #39/7#

Explanation:

The general term of an arithmetic sequence is given by the formula:

#a_n = a + d(n-1)#

where #a# is the initial term and #d# is the common difference.

The sum of #N# consecutive terms of an arithmetic sequence is #N# times the average of the first and last terms, so:

#sum_(n=1)^16 a_n = 16 (a_1 + a_16)/2 = 8(a+(a+15d)) = 16a+120d#

From the conditions of the question we have:

#240 = 16a + 120d#

#9 = a_5 = a+4d#

Hence:

#96 = 240 - 16*9 = (16a+120d) - 16(a+4d)#

#= color(red)(cancel(color(black)(16a)))+120d - color(red)(cancel(color(black)(16a)))-64d = 56d#

So #d = 96/56 = 12/7#

and #a = 9 - 4d = 9-48/7 = 15/7#

Hence the first three terms of the series are:

#15/7#, #27/7#, #39/7#