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Question

An arithmetic progression consists of 12 terms whose sum is 354.The ratio of the sum of the even terms to the sum of the odd terms is 32:27. Find the common difference of the progression.

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Solution

Sum of 12 terms=354
122[2a+(121)d]=3546[2a+11d]=3542a+11d=59(i)
SumofeventermsSumofoddterms=3227
Sum of even terms=32x & Sum of odd terms=27x
32x+27x=35459x=354x=6
Sum of even terms=32×6=192
t2+t4+t6+t8+t10+t12+192a1=t2=a+d(ii)
Common difference=t4t2=a+3dad=2d
Sum=192
n2[2a+(n1)d]=19262[2(a+d)+5×2d]=192[2a+2d+10d]=1923=642a+12d=64(iii)
Subtracting (i) from (iii)
d=5
The common difference=5.

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