Given sequence is 1,2,4,7,11,16... Find 125th term and also find sum.(good luck)
Clearly, the recursion relation for your sequence is
an+1 = an + n, with a1=1.
Since the n-th and (n+1)-th term differ by n, we suspect (from calculus) that the n-th term an has as its leading term n²/2. To ensure the leading term is divisible by 2, modify this as n(n-1)/2:
an=n(n-1)/2 + c for some unknown constant c.
After some trial and error, I found c=1, so
Therefore, the 125-th term is
a125=125*124/2+1 = 7751
Similarly, you find the partial sum
Again, calculus indicates that the leading term is of the form n³/6. After quite a bit of trial and error, I got
∑n=1125 an=(125³+5(125))/6 = 325,625.