{am} is a sequence of integers such that a1 = 1, and am+n = am + an + mn, for all positive integers m and n. What is the value of a11?

66

Step by Step Explanation:
1. We see that a1 = 1
a2 = a1 + a1 + 1 = 3
a3 = a2 + a1 + 2 = 6
a4 = a3 + a1 + 3 = 10 and so on ...
2. Seeing the sequence we can find out the pattern:
a4 = a3 + a1 + 3
or, a4 = a2 + a1 + 2 + a1 + 3 ...[Since a3 = a2 + a1 + 2]
or, a4 = a1 + a1 + 1 + a 1 + 2 + a1 + 3 ...[Since a2 = a1 + a1 + 1]
or, a4 = 4a1 + 1 + 2 + 3
or, a4 = (4 × 1) + 1 + 2 + 3 ...[Since a1 = 1]
or, a4 = 1 + 2 + 3 + 4
or a4 =
 4(4 + 1) 2
...[Since we know that the sum of n natural numbers is
 n(n+1) 2
]
3. Now, we conclude that am =
 m(m+1) 2

4. So the value of a11 is
 11 × (11 + 1) 2
= 66