2022-01-27
Suppose [tex]a_{n}[/tex] as the sum of the numbers in which the quotient and the remainder are equal when divided by [tex]n.[/tex]
For example, [tex]a_{5}=6+12+18+24=60.[/tex]
Find the least [tex]n[/tex] such that [tex]a_{n}\ \textgreater \ 500.[/tex]
Share
Step-by-step explanation:
Addition + Subtraction
1. Addition
The first trick is to simplify your problem by breaking it into smaller pieces. For example, we can rewrite
567 + 432
= 567 + (400 + 30 + 2)
= 967 + 30 + 2
= 997 + 2
= 999
Verified answer
We can observe general terms of n :
Rewrite it as :
a_n = (n + 1) + 2(n + 1) + 3(n + 1) + ... (n - 1)(n + 1)
= (n + 1)(1 + 2 + 3 + ... + n - 1)
= (n + 1)(1 + n - 1)(n -1)/2
= (n - 1)n(n + 1)/2
We need to find the least n for which a_n > 500 :
(n - 1)n(n + 1)/2 > 500
=> (n - 1)n(n + 1) > 1000
If n = 10,
If n = 11,
Therefore, The least n is 11.