n:1,2,3,4,5,6

f(n):0,1,3,6,10,15
find the nth term
and the 20th term in the sequence

for every letter n f(n) is added by the next number.

0,1
1,2
3,3
4,6,
5,10
6,15
7,21
8,28
9,34
10,43
11,53
12,64
13,76
14,89
15,103
16,118
17,134
18,151
19,169
20,188= answer

To find the nth term in the sequence, we can observe a pattern in the given sequence. The difference between consecutive terms in the f(n) sequence seems to be increasing by 1 with each term starting from 1 (1, 2, 3, 4, 5, ...). This indicates that we are dealing with a sequence of triangular numbers.

A triangular number is a number that can form an equilateral triangle with dots, where the dots are arranged in a triangular pattern. The formula to find the nth triangular number is:

T(n) = (n * (n + 1)) / 2

By plugging in the values of n from the given sequence into this formula, we can find the corresponding values of f(n):

For n = 1:
T(1) = (1 * (1 + 1)) / 2
= 2 / 2
= 1
So, f(1) = 0

For n = 2:
T(2) = (2 * (2 + 1)) / 2
= 6 / 2
= 3
So, f(2) = 1

For n = 3:
T(3) = (3 * (3 + 1)) / 2
= 12 / 2
= 6
So, f(3) = 3

For n = 4:
T(4) = (4 * (4 + 1)) / 2
= 20 / 2
= 10
So, f(4) = 6

For n = 5:
T(5) = (5 * (5 + 1)) / 2
= 30 / 2
= 15
So, f(5) = 10

Based on this pattern, we can deduce that the nth term of the sequence f(n) is equal to the (n - 1)th triangular number.

Now, to find the 20th term in the sequence, we need to substitute n = 20 into the formula for triangular numbers:

T(20) = (20 * (20 + 1)) / 2
= 420 / 2
= 210

Therefore, the 20th term in the sequence is f(20) = 210.

So, the nth term in the sequence is given by the (n - 1)th triangular number, and the 20th term is 210.