in an ap if S5 Plus S7 is equal to 167 and S10 is equal to 235 find AP where Sn denotes the sum of the first n terms

 S5 + S7 is equal to 167 and S10 is equal to 235 find AP where Sn denotes the sum of the first n terms

solution:-

 

if sn denotes the sum of first n terms of an AP then prove that s12=3(s8-s4)

solution 

LHS s12 = 12/2(2a+(12-1)d)

s12 = 6(2a+(11)d)

s12 = 12a+11×6d

 s12 = 12a+66d

RHS

=3(s8-s4)

=3{(8/2(2a+(8-1)d))-4/2(2a+(4-1)d)}

=3{4(2a+7d)-2(2a+3d)}

= 3{8a+28d-4a-6d)}

= 3{4a+22d}

=12a+66d

proved LHS=RHS 

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