Find the 20th term of 1/2,1/3,1/6,...........

Answer 1

Try to find the common difference of an arithmetic series. This particular series is the same as 3/6, 2/6, 1/6, ... So the common difference is 1/6.

a₁ = 3/6

a₂ = 3/6 - 1/6 = 2/6

a₃ = 3/6 - 1/6 - 1/6 = a₁ - 2(1/6) = a₁ - (3 - 1)(1/6)

a₄ = 3/6 - 1/6 - 1/6 - 1/6 = a₁ - 3(1/6) = a₁ - (4 - 1)(1/6)

Do you see the pattern here?

a_n = a₁ - (n - 1)/6

Now plug in n=20 and a₁=3/6.

Answer 2

there's more than one way....

1) multiply the denominators starting with 1/2 and 1/3:

So 1/2,1/3, 1/6, 1/18, 1/108, .....

2) denominators: add one, then double

So 1/2, 1/3, 1/6, 1/7, 1/14, 1/15, 1/30, 1/31, 1/62

Can we have the next term in the sequence to confirm, or rule these out?