{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?


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Answer:

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

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