Answer:
[tex]a_{32}=-173[/tex]
Step-by-step explanation:
Given : the arithmetic sequence where[tex]a_1 = 13[/tex] and [tex]a_{13} =-59[/tex]
We have to find the 32nd term of the arithmetic sequence.
Since, the general arithmetic sequence having first term 'a' and common difference 'd' is given by [tex]a_n=a+(n-1)d[/tex]
Thus, for the given arithmetic sequence, we have,
First term is [tex]a_1=a= 13[/tex] and
[tex]a_{13}=a+(13-1)d=-59[/tex]
Calculate the common difference by putting a = 13 in above, we have,
[tex]13+(12)d=-59[/tex]
Solving for d, we have,
[tex](12)d=-59-13[/tex]
[tex](12)d=-72[/tex]
Divide by 12 both side, we have,
[tex]d=-6[/tex]
Thus, the common difference is -6.
For 32nd term, Put a = 13 , d = -6 and n = 32 in [tex]a_n=a+(n-1)d[/tex]
We have,
[tex]a_{32}=13+(32-1)(-6)[/tex]
Simplify, we have,
[tex]a_{32}=13+(31)(-6)[/tex]
[tex]a_{32}=13-186[/tex]
[tex]a_{32}=-173[/tex]
