For the sequence 0,3,8,15,24,35....
find a formula for tsubn in terms of n.
the differences are 3,5,7,9,11
the sum of the first n odd numbers is n^2, so that gives us a clue
Tn = n^2 - 1
To find a formula for the nth term (tsubn) of the given sequence, we need to observe and identify the pattern in the sequence.
Looking at the sequence 0, 3, 8, 15, 24, 35..., we can notice that each term is obtained by adding a consecutive odd number to the previous term. Let's break it down further:
0 + 3 = 3
3 + 5 = 8
8 + 7 = 15
15 + 9 = 24
24 + 11 = 35
...
We can see that the consecutive odd numbers are 3, 5, 7, 9, 11,...
To find the nth term (tsubn), we can set up a formula using this pattern. The first term of the sequence is when n = 1, which is 0.
Formula: tsubn = (n-1)^2 + 1
Let's verify this formula for a few terms:
When n = 1: tsub1 = (1-1)^2 + 1 = 0 + 1 = 1 (first term of the sequence)
When n = 2: tsub2 = (2-1)^2 + 1 = 1 + 1 = 2
When n = 3: tsub3 = (3-1)^2 + 1 = 4 + 1 = 5
When n = 4: tsub4 = (4-1)^2 + 1 = 9 + 1 = 10
Therefore, the formula tsubn = (n-1)^2 + 1 gives us the nth term of the sequence 0, 3, 8, 15, 24, 35,...