The 10th term of an arithmetic sequence is 10 and the sum of the first 10 terms is -35. What is the first term and the common difference of the sequence?

1 Answer
sente
Nov 27, 2015

The first term is #-17# and the common difference is #3#.

Explanation:

The sum of the first #n# terms of an arithmetic sequence #a_1, a_2, a_3, ...# with first term #a_1# and common difference #d#
(i.e. #a_k = a_1+d(k-1)#)
is given by

#sum_(k=1)^na_k = n((a_1+a_n)/2)#

As the sum of the first ten terms is #-35# that gives us

#10((a_1+a_10)/2) = -35#

As the tenth term is #10# that gives us

#10((a_1+10)/2) = -35#

#=> a_1+10 = -7#

#=> a_1 = -17#

Which gives us the first term as #-17#. To find the difference, we note that
#10 = a_10 = a_1+9d = -17+9d#

#=> 9d = 27#

#=> d = 3#

Thus the first term is #-17# and the common difference is #3#.

Related questions
Impact of this question
12647 views around the world
You can reuse this answer
Creative Commons License