For the sequence 3,8,15,24,35,48,63
write an equation for t(subn) in terms of n.
the differences are 5,7,9,11,...
since n^2 is the sum of the first n odd numbers, that suggests we look for a square in here.
Tn = (n+1)^2 - 1
To find an equation for t(subn) in terms of n, we need to identify the pattern in the sequence. By observing the differences between consecutive terms, we notice that the differences are increasing by 5 each time.
Here are the differences between consecutive terms:
8 - 3 = 5
15 - 8 = 7
24 - 15 = 9
35 - 24 = 11
48 - 35 = 13
63 - 48 = 15
As the differences are increasing by 5 each time, we can express this difference as (n - 1) * 5.
Now, let's find the equation for t(subn):
t(subn) = t(1) + (n - 1) * 5
Substituting the value of t(1) (which is the first term of the sequence), we get:
t(subn) = 3 + (n - 1) * 5
Therefore, the equation for t(subn) in terms of n is t(subn) = 3 + (n - 1) * 5.
To write an equation for t(subn) in terms of n, we need to find a pattern in the given sequence.
By observing the sequence, we can see that each term is obtained by adding consecutive odd numbers.
Let's break it down:
- The first term, 3, is obtained by adding 1 (1st odd number) to 2 (the previous number).
- The second term, 8, is obtained by adding 3 (2nd odd number) to 5 (the previous number).
- The third term, 15, is obtained by adding 5 (3rd odd number) to 10 (the previous number).
- The fourth term, 24, is obtained by adding 7 (4th odd number) to 17 (the previous number).
- And so on...
From the pattern, we can conclude that to obtain the nth term, we need to find the sum of the first n odd numbers and add it to (n-1)².
The formula for the sum of the first n odd numbers is given by: n².
Therefore, the equation for t(subn) can be written as:
t(subn) = n² + (n-1)²
In this equation, replacing n with the position of the term will give you the value of that term in the sequence.