Integration

Dear student

Consider (x+y)ⁿ   general term = ⁿC_r X^n-r Y ^r

(  (7)^1/2  +  (5)^1/6  )^78 in this case general term =⁷⁸C_r 7 ^(78-r)/2 5^r/6

To get integer terms  (78-r)/2  and r/6 must be integers.

Consider (78-r)/2 will be integer if and only if  r is even , so allowed r values are  0, 2,4 ......78.

r/6   will be integer if and only if  r is a multiple of  6, so allowed r values are  0,6 , 12, ...78.

See the common values of r in above to series of values.

So there will be  14 intigral terms .

Simple way to find  solution is lake LCM of 2 and 6 . Devide the 78 by that LCM and add 1 to the answer ( 0 th term)

Regards

Topperlearning team.