seqeunces (1 Viewer)

stressedadfff · Aug 12, 2021

hey guys does anyone know how to find "n" for iii?

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stressedadfff · Aug 12, 2021

and also c pls

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CM_Tutor · Aug 12, 2021

stressedadfff said:
hey guys does anyone know how to find "n" for iii?
View attachment 31588

The series $100 + 97 + 94 + ...$ is an AP with $a = 100$ and $d = -3$ . Its partial sum is

$S_n = \cfrac{n}{2}\bigg[2a + (n - 1)d\bigg] = \cfrac{n}{2}\bigg[2(100) - 3(n - 1)\bigg] = \cfrac{n}{2}(203 - 3n)$

For the partial sum to be negative, put $S_n < 0$ :

$\begin{align*} S_n &< 0 \\ \cfrac{n}{2}(203 - 3n) &< 0 \\ n(203 - 3n) &< 0 \end{align*}$

The graph of $y = f(n)$ for the function $f(n) = n(203 - 3n)$ is a concave down parabola with roots at $n = 0$ and $203 - 3n = 0 \implies n = \cfrac{203}{3} = 67\cfrac{2}{3}$ .

So, the solution of $f(n) < 0$ is $n < 0$ and $n > 67\cfrac{2}{3}$ .

However, since $n$ is a number of terms, it must be a positive integer, and so the solution of $S_n < 0$ is $n \geqslant 68$ .

It follows that the smallest number of terms for which $S_n < 0$ is $n = 68$ .

stressedadfff · Aug 12, 2021

thank you! cwould you be able to help with the previous please

CM_Tutor · Aug 12, 2021

stressedadfff said:
and also c pls
View attachment 31591

Lawrence's wages each year are $50 000, $55 000, $60 000, and these are the first three terms in an AP with first term $50 000 and common difference $5 000. If his wages are $50 000 in year 1 and $55 000 in year 2, then in year $n$ his wage will be

$L_n = a + (n - 1)d = 50 000 + 5000(n - 1) = 5000n + 45 000$

Julian's wages each year are $40 000, $46 000, $52 900, and these are the first three terms in a GP with first term $40 000 and common ratio 1.15. If his wages are $40 000 in year 1 and $46 000 in year 2, then in year $n$ his wage will be

$J_n = ar^{n - 1} = 40 000(1.15)^{n-1}$

Julian's wage exceeds Lawrence's when $J_n > L_n$ or when the difference in year $n$ ( $D_n$ ), given by $D_n = J_n - L_n$ is positive.

If you test the values of $n$ , you'll find $D_n < 0$ for $n = 1,\ 2,\ ...,\ 5$ and $D_n > 0$ for $n \geqslant 6$

stressedadfff · Aug 13, 2021

thank you! could someone please send a solution for the following:

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