a) To find an expression for Sn-1 in terms of n, we substitute (n-1) for n in the original expression Sn = n^2 + 2n.
So, we have:
Sn-1 = (n-1)^2 + 2(n-1)
Expanding the squared term:
Sn-1 = (n^2 - 2n + 1) + 2(n-1)
Simplifying further:
Sn-1 = n^2 - 2n + 1 + 2n - 2
Combining like terms, we get:
Sn-1 = n^2 - 1
Therefore, the expression for Sn-1 in terms of n is n^2 - 1.
b) To find an expression for the nth term, tn, in terms of n, we can subtract Sn-1 from Sn.
So, tn = Sn - Sn-1
Substituting the expressions we found earlier:
tn = (n^2 + 2n) - (n^2 - 1)
Expanding and simplifying:
tn = n^2 + 2n - n^2 + 1
Combining like terms, we get:
tn = 2n + 1
Therefore, the expression for the nth term, tn, in terms of n is 2n + 1.