The 8
th
term of an arithmetic progression is −17, and the 17
th
term is −71. Find the 13
th
term.

Let the first term of the arithmetic progression be a and the common difference be d.
The formula for the n
th
term of an AP is:

t
n
​
=a+(n−1)d
Given information:

8
th
term (t
8
​
) is −17:

a+7d=−17— (Equation 1)
17
th
term (t
17
​
) is −71:

a+16d=−71— (Equation 2)
Subtract Equation 1 from Equation 2:

(a+16d)−(a+7d)=−71−(−17)
9d=−71+17
9d=−54
d=−6
Substitute d=−6 into Equation 1 to find a:

a+7(−6)=−17
a−42=−17
a=−17+42
a=25
Now, find the 13
th
term (t
13
​
):

t
13
​
=a+12d
t
13
​
=25+12(−6)
t
13
​
=25−72
t
13
​
=−47