SOLUTION: Given an arithmetic sequence with total of the first three terms is equal to 13 and the last N th term is equal to 7. Determine the middle term of arithmetic sequence and its value

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Question 937656: Given an arithmetic sequence with total of the first three terms is equal to 13 and the last N th term is equal to 7. Determine the middle term of arithmetic sequence and its value
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Any term at index i would be A%5B1%5D%2B%28i-1%29d, for common difference of d.
Your description with the last N th term must mean, index of 3 because your sequence is given to have three terms.

To be simple, a=A%5B1%5D.

a%2B%281-1%29d%2Ba%2B%282-1%29d%2Ba%2B%283-1%29d=13
a%2Ba%2Bd%2Ba%2B2d=13
highlight_green%283a%2B3d=13%29
One equation in two unknowns.

More from the description,
a%2B%283-1%29d=7
highlight_green%28a%2B2d=7%29

Now, we have two equation in the two unknowns, a and d.

system%283a%2B3d=13%2Ca%2B2d=7%29

system%283a%2B3d=13%2C3a%2B6d=21%29, starting with Elimination Method
E2-E1 gives.....

3d=8
highlight%28d=8%2F3%29

The middle, or term at index 2, is 7-8%2F3=21%2F3-8%2F3=highlight%2813%2F3%29.


....Only three terms are in the sequence.
"...sequence with total of the first three terms is equal to 13..."
and
" last N th term is equal to 7."
and
"Determine the middle term of arithmetic sequence".
Those strongly imply that the last term of the sequence is at index number 3.