Question
For which positive integers n is the ratio, 
an integer?
- odd n only
- even n only
- n = 1 + 6k only, where k
0 and k
I
- n = 1 + 3k, integer k 0
0 and kI
0
Hint:
find the simple expression of the given expression and find integer values of n from the resultant expression that satisfy the condition.
The correct answer is: n = 1 + 3k, integer k 0
n= 1 + 3k, k>=0 k c I
on solving the given expression, we get
(n(n+1)(2n+1)/6)/(n(n+1)/2)
= (2n+1)/3
the above expression needs to be an integer for integer values of n
let this be p
therefore, 2n+1 = 3p
n=(3p-1)/2
since p belongs to the set of integers, lets put values of p
p=1 : n= 1 ,
p=2: n= 5/2
p=3: n= 4
p=4: n= 11/2
p=5: n= 7
p= 6 : n= 17/2
hence, we can see that the values satisfying the condition are:
n={1,3,7,10,...}
this can be generated from the equaiton
n= 1+3k, integer k>=0
this is our answer.
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements. the variable n is generalized using an AP as well.
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