### {a_{m}} is a sequence of integers such that a_{1} = 1, and a_{m+n} = a_{m} + a_{n} + mn, for all positive integers m and n. What is the value of a_{11}?

**Answer:**

66

**Step by Step Explanation:**

- We see that a
_{1 }= 1

a_{2}= a_{1 }+ a_{1 }+ 1 = 3

a_{3}= a_{2}+ a_{1 }+ 2 = 6

a_{4}= a_{3}+ a_{1 }+ 3 = 10 and so on ... - Seeing the sequence we can find out the pattern:

a_{4}= a_{3}+ a_{1}+ 3

or, a_{4}= a_{2}+ a_{1}+ 2 + a_{1}+ 3**...[Since a**_{3}= a_{2}+ a_{1}+ 2]

or, a_{4}= a_{1 }+ a_{1 }+ 1 + a_{ 1 }+ 2 + a_{1 }+ 3**...[Since a**_{2}= a_{1 }+ a_{1}+ 1]

or, a_{4}= 4a_{1}+ 1 + 2 + 3

or, a_{4}= (4 × 1) + 1 + 2 + 3**...[Since a**_{1}= 1]

or, a_{4}= 1 + 2 + 3 + 4

or a_{4}=4(4 + 1) 2 **...[Since we know that the sum of n natural numbers is**

]n(n+1) 2 - Now, we conclude that a
_{m}=m(m+1) 2 - So the value of a
_{11}is

= 6611 × (11 + 1) 2