Geometriske rekker - eksempel 2

Oppgave 2

Anta at $\displaystyle \sum_{n=1}^{\infty} (a_n+2 b_n) = 3$ og $\displaystyle \sum_{n=1}^{\infty} b_n = 5$. Finn $\displaystyle \sum_{n=1}^{\infty} a_n$.

Løsning

Vi bruker at $\displaystyle \sum_{n=1}^{\infty}(a_n+b_n) = \sum_{n=1}^{\infty}a_n + \sum_{n=1}^{\infty}b_n$ og $\displaystyle \sum_{n=1}^{\infty}(c a_n) = c\sum_{n=1}^{\infty}a_n$, og får:

\begin{align*} \sum_{n=1}^{\infty} (a_n+2 b_n) &= 3 \\ \sum_{n=1}^{\infty} a_n+ \sum_{n=1}^{\infty}2 b_n &= 3 \\ \sum_{n=1}^{\infty} a_n+ 2\sum_{n=1}^{\infty}b_n &= 3 \\ \sum_{n=1}^{\infty} a_n+ 2\cdot 5 &= 3 \\ \sum_{n=1}^{\infty} a_n+ 10 &= 3 \\ \sum_{n=1}^{\infty} a_n &= \underline{\underline{-7}} \\ \end{align*}