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# In an A.P. (with usual notations): (i) given a=5,d=3,a n ​ =50, find n and S n ​ (ii) given a=7,a 13 ​ =35, find d and S 13 ​ (iii) given d=5,S 9 ​ =75, find a and a 9 ​ (iv) given a=8,a n ​ =62,S n ​ =210, find n and d (v) given a=3,n=8,S 8 ​ =192, find d

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It is given that In an A.P. (with usual notations):
(i) given a=5,d=3,an=50, we have asked to find n and Sn
(ii) given a=7,a13=35, we have been asked to calculate  d and S13
(iii) given d=5,S9=75, we have been asked to determine  a and a9
(iv) given a=8,an=62,Sn=210, we have been asked to find n and d
(v) given a=3,n=8,S8=192, calculate  d

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### 1 Answer

1. (i) From the question,
First term a=5
Then common difference d=3
an=50
We know that, an=a+(n1)d
50=5+(n1)350=5+3n350=2+3n3n=5023n=48n=48/3
n=16So,Sn=====(n/2)(2a+(n1)d)(16/2)((2×5)+(161)×3)8(10+45)8(55)440

(ii) given a=7,a13=35, find d and S13
From the question,
First term a=7
a13=35
We know that, an=a+(n1)d
35=7+(131)d35=7+13dd35=7+12d12d=35712d=28d=28/12
d=7/3
So,S13=(n/2)(2a+(n1)d)=(13/2)((2×7)+((131)×(7/3)))=(13/2)((14+(12×7/3))=(13/2)(14+28)=(13/2)(42)=13×21=273

(iii) given d=5,S9=75, find a and a9
From the question it is given that,
Common difference d=5
S9=75
We know that, an=a+(n1)d
a9=a+(91)5a9=a+455a9=a+40
Then, S9=(n/2)(2a+(n1)d)75=(9/2)(2a+(91)5)75=(9/2)(2a+(8)5)(75×2)/9=2a+40150/9=2a+402a=150/9402a=50/3402a=(50120)/32a=−70/3a=−70/(3×2)a=−35/3
Now, substitute the value of a in equation (i),
a9=a+40=−35/3+40=(−35+120)/3=85/3

(iv) given a=8,an=62,Sn=210, find n and d
From the question, it is given that,
First-term a=8
an=62 and Sn=210
We know that, an=a+(n1)d
62=8+(n1)d(n1)d=628(n1)d=54 Then, Sn=(n/2)(2a+(n1)d)210=(n/2)((2×8)+54)210=(n/2)(16+54)420=n(70)n=420/70n=6
Now, substitute the value of n in equation (i),
(n1)d=54(61)d=545d=54d=54/5
Therefore, d=54/5 and n = 6

(v) given a=3,n=8,S8=192, find d
From the question, it is given that,
First term a=3
n=8S8=192
We know that, Sn=(n/2)(2a+(n1)d)
192=(8/2)((2×3)+(81)d)192=4(6+7d)192/4=6+7d48=6+7d486=7d42=7dd=42/7d=6
Therefore, the common difference d is 6.

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