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Question

In an A.P
i) Given a=2,d=8,Sn=90, Find n & a9
ii) Given d=5,S9=75, find a&a9
iii) Given a=7,a13=35, find d&S13

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Solution

(i)
Here a=2,d=8,Sn=90

We know Sn=n2[2a+(n1)d]

90=n2[22+(n1)8]

180=n[4+8n8]

180=8n24n

8n24n180=0

2n2n45=0

(n5)(2n+9)=0

n=5,92

n=5 (since n cannot be negative)

Therefore an=a+4d=2+4(8)=2+32=34

(ii)

Here d=5,S9=75

We know Sn=n2[2a+(n1)d]

S9=92[2a+(8)d]

75=92[2a+8(5)]

75=92[2a+40]

75=9[a+20]

75=9a+180

9a=105

a=353

Therefore a9=a+8d=353+8(5)=853

(iii)

Here a=7,a13=35

We know an=a+(n1)d

a13=7+(131)d

35=7+12d

28=12d

d=73

We know Sn=n2[2a+(n1)d]

S13=132[2(7)+(131)73]

=132[14+12×73]

=132[14+28]

=132(42)

=13×21

S13=273


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