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Question

In the A.P 1,7,13,19,....415 prove that the sum of the middle terms is equal to the sum of first and last terms.

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Solution

We observe that 1,7,13,19,,415 is an A.P. with first term a=1 and common difference d=6. Let there be n terms in the given A.P.

Then,
nth term 415
a+(n1)d=415
1+6(n1)=415
6n=420
n=70

So, there are 70 terms in the given A.P.
Therefore, (702)th=35th and (702+1)th=36th are the middle terms.

Now,
a35+a36=a+34d+a+35d=1+34×6+1+35×6=205+211=416

Also, a1+a70=1+415=416
Clearly, a35+a36=a1+a70.

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