Find the number of terms in the series `20 ,19 1/3,18 2/3...` the sum of which is 300. Explain the answer.

Explanation

The given sequence is an A.P. with first term a=20 and the common difference d=-2/3. Let the sum of n terms be 300. Then,
`S_(n)=300`
or `n/2[2a+(n-1)d]=300`
or `n/2[2xx20+(n-1)(-2//3)]=300`
or `n^(2)-61n+900=0`
or (n-25)(n-36)=0
`rArrn=25or36`
So, the sum of 25 terms is equal to the sum of 36 terms, which is equal to 300.
Here the common difference is negative, therefore terms go on diminishing and the `31^(st)` term becomes zero. All terms after the `31^(st)` term are negative. These negative terms when added to positive terms from `26^(th)` term to `30^(th)` term, cancel out each other and the sum remains same. Hence, the sum of 25 terms as well as that of 36 terms is 300.