a) To determine if the sequence is monotone, we need to check if an+1 > an for all n. We have:
a_n+1 = 3^(n+1) + 2^(n+1) - 2*2^n = 3*3^n + 2*2^n - 2*2^n = 3*3^n
a_n = 3^n + 2^n - 2*2^(n-1) = 3^n - 2^n
So we need to check if 3*3^n > 3^n - 2^n for all n. This simplifies to 2^n > 0, which is true for all n. Therefore, the sequence is monotone.
To determine if the sequence is bounded, we can find its limit as n approaches infinity. We have:
lim(n->inf) an = lim(n->inf) (3^n + 2^n - 2*2^(n-1))
= lim(n->inf) 3^n + lim(n->inf) 2^n - lim(n->inf) 2*2^(n-1)
= inf + inf - inf = undefined
Since the limit does not exist, the sequence is not bounded.
b) The given series is a geometric series with first term a = 15 and ratio r = -3/5. To determine if it is convergent, we need to check if |r| < 1. Since |r| = 3/5 < 1, the series is convergent.
The sum of a convergent geometric series is given by:
S = a/(1-r)
Plugging in the values, we get:
S = 15/(1-(-3/5)) = 15/(8/5) = 93.75
Therefore, the sum of the series is 93.75.