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If 10 times the 10thterm of an A.P. is equal to 15 times the 15th term, show that the 25thterm of A.P. is zero.

Answer
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Hint: Use general term of A.P. i.e, Tn=a+(n1)d.
We, know that the nthterm of an A.P. is given as:
Tn=a+(n1)d
10thterm of A.P. will be:
T10=a+(101)dT10=a+9d
Similarly, 15thterm will be:
T15=a+(151)d,T15=a+14d
Now, according to question:
10T10=15T15
So, putting values of T10and T15from above, we’ll get:
10(a+9d)=15(a+14d)10a+90d=15a+210d5a+120d=0a+24d=0
And 25thterm of A.P. will be:
T25=a+(251)dT25=a+24d
Putting the value a+24d=0 from above, we get:
T25=0.
Hence the 25thterm of A.P. is zero.
Note: Since 25th term of A.P. is zero, we can conclude that the sum of the first 49 terms of this A.P. is zero. In that case, the sum of the first 24 terms will be negative of the sum of the last 24 terms and 25th term is already zero.
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