The sum of the first $n$ terms of a series, ${{S}_{n}}$, is given by $\frac{{{a}^{n}}}{{{2}^{n-1}}}-

Question

The sum of the first $n$ terms of a series, ${{S}_{n}}$, is given by $\frac{{{a}^{n}}}{{{2}^{n-1}}}-2$, where $a$ is a non-zero constant and $a
e 2$. Show that ${{T}_{n}}$, the ${{n}^{\mathrm{th}}}$ term of the series, is $\left( a-2 ight){{\left( \frac{a}{2} ight)}^{n-1}}$. Hence show that the given series is a geometric series. [4] Find the range of values of $a$ for the sum to infinity to exist. [2] Given that $a=1$, find the least value of $n$ for ${{S}_{n}}$ to be within $\pm 0.2$ of the value of the sum to infinity. [3]

Accepted Answer

(i) geometric with common ratio $\frac a2$; (ii) $-2<a<2$; (iii) least $n=4$

2021 SAJC Revision Package Q2

2021 SAJC Revision Package Q2

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