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

# 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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(i) From the question,

First term a=5

Then common difference d=3

an=50

We know that, an=a+(n−1)d

50=5+(n−1)350=5+3n−350=2+3n3n=50−23n=48n=48/3

n=16So,Sn=====(n/2)(2a+(n−1)d)(16/2)((2×5)+(16−1)×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+(n−1)d

35=7+(13−1)d35=7+13d−d35=7+12d12d=35−712d=28d=28/12

d=7/3

So,S13=(n/2)(2a+(n−1)d)=(13/2)((2×7)+((13−1)×(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+(n−1)d

a9=a+(9−1)5a9=a+45−5a9=a+40

Then, S9=(n/2)(2a+(n−1)d)75=(9/2)(2a+(9−1)5)75=(9/2)(2a+(8)5)(75×2)/9=2a+40150/9=2a+402a=150/9−402a=50/3−402a=(50−120)/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+(n−1)d

62=8+(n−1)d(n−1)d=62−8(n−1)d=54 Then, Sn=(n/2)(2a+(n−1)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),

(n−1)d=54(6−1)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+(n−1)d)

192=(8/2)((2×3)+(8−1)d)192=4(6+7d)192/4=6+7d48=6+7d48−6=7d42=7dd=42/7d=6

Therefore, the common difference d is 6.