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The number of integral terms in the expansion of (7 1/2 +5 1/6 ) 78 is a) 15 b) 14 c) 16 d) 0 I know the ans is b. But I want the method of solving it. What is the condition for finding the no. of integral terms?

The number of integral terms in the expansion of (71/2+51/6)78 is


a) 15


b) 14


c) 16


d) 0


I know the ans is b. But I want the method of solving it. What is the condition for finding the no. of integral terms?


 

Grade:Upto college level

4 Answers

askIITIians Expert
21 Points
15 years ago

Dear Sohini,

This is a good question; a lay man approach has been presented for this problem.

You have,

(71/2 + 51/6 )78        

Now, general term of this binomial will be

Tr+1 = 78Cr  (71/2)78-r    (51/6)r

This general term will be an integer if 78Cr  is an integer and 778-r/2 is an integer and 5r/6 is an integer

78Cr will always be a +ve integer

Since 78Cr denotes no. of ways of selecting r things out of 78 things, it cannot be a fraction.

778-r/2  will be an integer

If  78-r/2 is an integer

r = 0 , 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16, ………………..70 , 72 , 74 , 76 and 78                         (i)

Also 5 is an integer if r/6 is an integer

r = 0 , 6, 12 , 18 , 24 , …………………                                    (ii)

Taking the intersection (i) and (ii)

We have,

r = 0 , 6 , 12 , 18 , 24 , 30 , 36 , 42 , 48 , 54, 60 , 66 , 72 , 78

Hence total r permissible is 14

Therefore you have 14 terms

 

Siddharth Singh
22 Points
9 years ago
What if the degree is high ? 
Like here.... Integrals solutions in expansion of (5^1/2 + 7^1/8)^1024 ? 
deepti
15 Points
5 years ago
we can use this simple formula for finding integral co efficients ,no of rational nos...
n/L.C.M(p,q) whr p,q are the nos..eg 
lets solve tht question itself
78/lcm(2,6)=78/6=13 so to find the tot terms we do n+1 so 13+1=14 terms
another eg 1024/lcm(2,8)=128 ,tot terms =129
ankit singh
askIITians Faculty 614 Points
3 years ago

From binomial theory we know that the (r+1)th term of the expansion

(a+b)^n is (nCr)*(a^n-r)*(b^r)

In this case a = 5^(1/6) ; b = 2^(1/8) ; n=100 ;

=> Total number of terms = n+1 = 101

For a term to be rational the powers of ‘a’ and ‘b’ should be integral multiples of 6 and 8 respectively so as to cancel out the fractional exponents.

=>n-r = 6k and r = 8m where k and m are some non negative integers

=> r should take values 0,8,16,24,32,40,48,56,64,72,80,88,96 (8m)

=> n-r should take values among 0,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96 (6k)

As n= 100 for n-r=6k, r should take 100,94,88,82,76,70,64,58,52,46,40,34,28,22,16,10,4

The common values of r which satisfy r = 8m and n-r = 6k simultaneously are r = 16,40,64,88

=> There will be 4 rational terms

=> The remaining 97 terms will be irrational

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