For the sequence 3,8,15,24,35,48,63
write an equation for t(subn) in terms of n.
To find an equation for the sequence 3, 8, 15, 24, 35, 48, 63 in terms of n, we need to observe the pattern of the sequence. Let's start by finding the differences between consecutive terms:
8 - 3 = 5
15 - 8 = 7
24 - 15 = 9
35 - 24 = 11
48 - 35 = 13
63 - 48 = 15
We notice that the differences between consecutive terms are increasing by 2 each time. This suggests that the sequence may be quadratic, and we can try to find an equation of the form t(subn) = an^2 + bn + c.
To determine the values of a, b, and c, let's expand the equation:
t(subn) = an^2 + bn + c
Substituting n = 1:
3 = a(1)^2 + b(1) + c
3 = a + b + c ...(1)
Substituting n = 2:
8 = a(2)^2 + b(2) + c
8 = 4a + 2b + c ...(2)
Substituting n = 3:
15 = a(3)^2 + b(3) + c
15 = 9a + 3b + c ...(3)
Solving this system of equations can help us find the values of a, b, and c.