Question

Can someone please explain to me how the formula Sn=n(t1+tn/2) works?

Answer 1

this?

\[S_n = \frac{n}{2}(t_1 + t_n)\]

well this is the sum to n of an arithmetic sequence

lets consider the following arith sequence, common difference d

\[t_1 \text{ , } t_1 + d \text{ , } t_1 + 2d \text{ , ... , }t_1 + (n-1)d\]
note that the last term ( t_n) is:
\[t_1 + (n-1)d\]
\[S_n = t_1 + (t_1+d) + (t_1+2d) + ... (t_1 + (n-2))d+ (t_1 +(n-1)d)\]
reversed it looks like this:
\[S_n = (t_1 +(n-1)d) +(t_1+(n-2)d)+ ... + (t_1+2d) + (t_1 +d) +t_1\]

imagine adding these two together, by adding the first term in the first sum to the last term in the second sum,



using the fact that
\[t_n = t_1 + (n-1)d\]
we are left with your formula.