<p>The sum of the first $n$ terms of a sequence, $G$, is given by ${{S}_{n}}=c\left( 2-{{2}^{1-n}} \right)$, where $c$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the ${{n}^{\mathrm{th}}}$ term of the sequence $G$, ${{T}_{n}}=c{{2}^{1-n}}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, prove that the sequence of $G$ is a geometric progression.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that the sixth term of $G$ is $\frac{3}{8}$. Show that ${{T}_{1}}=12$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The first term and common difference of another arithmetic progression $A$ are the same as the first term and the common ratio of $G$ respectively. Find the least value of $n$ such that the sum of the first $n$ terms of the arithmetic progression $A$ is more than one-quarter the sum of the first $n$ terms of $G$ by at least $250$.</span></div><p style="text-align: right;">[3]</p></li></ol>

The sum of the first $n$ terms of a sequence, $G$, is given by ${{S}_{n}}=c\left( 2-{{2}^{1-n}} \right)$, where $c$ is a positive constant.

Show that the ${{n}^{\mathrm{th}}}$ term of the sequence $G$, ${{T}_{n}}=c{{2}^{1-n}}$.

Hence, prove that the sequence of $G$ is a geometric progression.

It is given that the sixth term of $G$ is $\frac{3}{8}$. Show that ${{T}_{1}}=12$.

The first term and common difference of another arithmetic progression $A$ are the same as the first term and the common ratio of $G$ respectively. Find the least value of $n$ such that the sum of the first $n$ terms of the arithmetic progression $A$ is more than one-quarter the sum of the first $n$ terms of $G$ by at least $250$.

2020 SAJC Promo Q6

2020 SAJC Promo Q6

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