How many terms of the Arithmetic Progressions 78,71,64,.........are needed for the sum to give the sum 465? Also find the last term of the AP?
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78,71,64,....are in AP
first term =a =78
common difference =d =a2-a1=71-78=-7
1) sum of n terms = sn=465
n/2[2*78+(n-1)(-7)]=465
n(156-7n+7) = 930
156n-7n²+7n=930
-7n²+163n=930
7n²-163n-930=0
7n²-70n+93n-930=0
7n(n-10)+93(n-10)=0
(n-10)(7n+93)=0
therefore
n-10=0 or 7n+93=0
n value should not be a fraction
n-10= 0
n= 10
sum of 10 terms =465
2)last term = an = a+(n-1)d
= 78+(n-1)(-7)
=78-7n+7
=85-7n
first term =a =78
common difference =d =a2-a1=71-78=-7
1) sum of n terms = sn=465
n/2[2*78+(n-1)(-7)]=465
n(156-7n+7) = 930
156n-7n²+7n=930
-7n²+163n=930
7n²-163n-930=0
7n²-70n+93n-930=0
7n(n-10)+93(n-10)=0
(n-10)(7n+93)=0
therefore
n-10=0 or 7n+93=0
n value should not be a fraction
n-10= 0
n= 10
sum of 10 terms =465
2)last term = an = a+(n-1)d
= 78+(n-1)(-7)
=78-7n+7
=85-7n
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