The sum of the first $n$ terms of a sequence, ${{S}_{n}}$ , is a quadratic polynomial in $n$.

Question

The sum of the first $n$ terms of a sequence, ${{S}_{n}}$ , is a quadratic polynomial in $n$. It is given that the first term and second term of the sequence is $50$ and $45$ respectively, and that the sum of the next $10$ terms is $175$. Find ${{S}_{n}}$ and hence show that this sequence is an arithmetic progression.[5] At the end of 2015, the farmers of a chocolate factory had $40000$ square metres of fertile land for planting cocoa. At the end of 2016, the farmers realised that the area of fertile land had decreased by 800 square metres due to nutrient loss from the soil. In each subsequent year, the decrease in the area of fertile land is $\frac{7}{8}$ of the decrease in the previous year. Show that the remaining area of fertile land at the end of the ${{n}^{\mathrm{th}}}$ year starting from 2016 (where 2016 is the 1st year, 2017 is the 2nd year, and so on) can be expressed as $a+b{{\left( \frac{7}{8} \right)}^{n}}$, where $a$ and $b$ are constants to be determined.[3] After how many complete years, starting from 2016, will the area of fertile land first fall below $35000$ square metres? Show your working clearly.[3] Find the area of fertile land available to the farmer in the long run.[1]

Accepted Answer

(a) ${{S}_{n}}=-\frac{5}{2}{{n}^{2}}+\frac{105}{2}n$ (b)(i) $a=33600,b=6400$ (ii) $12$ (iii) $33\,600$ square metres

2017 DHS Promo Q11

2017 DHS Promo Q11

Video Solution:

Solution:

Solution locked