how do i solve if t1=1/2 and tn=tn-1+1/n(n+1), determine an explicit formula for the general term. (hint: it is neither arithmetic nor geometric. write out the first few terms)

1) 1/2

2) 1/2 + 1/(2*3) = 1/2 + 1/6 = 2/3

3) 2/3 + 1/(3*4) = 2/3 + 1/12 = 3/4

4) 3/4 + 1/(4*5) = 15/20 + 1/20 = 4/5

LOL - See a pattern?

To find an explicit formula for the general term of the sequence, we first need to observe the given recursive formula and determine the pattern by writing out the first few terms.

Given:
t₁ = 1/2
tₙ = tₙ₋₁ + 1/n(n+1)

To find the explicit formula, let's calculate the first few terms of the sequence:

t₁ = 1/2
t₂ = t₁ + 1/2(2+1) = 1/2 + 1/2(3) = 1/2 + 3/2 = 4/2 = 2
t₃ = t₂ + 1/3(3+1) = 2 + 1/3(4) = 2 + 4/3 = 6/3 + 4/3 = 10/3
t₄ = t₃ + 1/4(4+1) = 10/3 + 1/4(5) = 10/3 + 5/4 = 40/12 + 15/12 = 55/12

From the first few terms, we can observe a pattern in the sequence:

t₁ = 1/2
t₂ = 2
t₃ = 10/3
t₄ = 55/12

To find the explicit formula, we can notice that the numerators of the terms are increasing, while the denominators are multiplying with consecutive integers. So, we can express the general term as:

tₙ = Aₙ/Bₙ

where Aₙ is the sequence of numerators and Bₙ is the sequence of denominators.

Looking at the pattern, we can identify that the numerators follow the sequence:

A₁ = 1, A₂ = 2, A₃ = 10, A₄ = 55

and the denominators follow the sequence:

B₁ = 2, B₂ = 3, B₃ = 4, B₄ = 5

To express these sequences as functions, we can use polynomial interpolation or observe the pattern:

Aₙ = (n-1)²(n+2)
Bₙ = n+1

Now we can rewrite the general term as follows:

tₙ = Aₙ/Bₙ = (n-1)²(n+2) / (n+1)

Therefore, the explicit formula for the general term of the sequence is:

tₙ = (n-1)²(n+2) / (n+1)