# Series :) enjoy

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• November 12th 2012, 07:37 AM
Redwood87
Series :) enjoy
1) Water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second, 550 during the third and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m.

2) A firm starts work with 110 employees for the 1st week. the number of the employees rises by 6% for the first week. how many persons will be employed in the 20th week if the present rate of expansion continues.

3) a contractor hires out machinery. in the first year of hiring out one piece of equipment the profit is $6000, but this diminishes by 5% in successive years. show that the annual profits form a geometric progression and find the total of all the profits for the first 5 years. • November 12th 2012, 07:43 AM Prove It Re: Series :) enjoy Quote: Originally Posted by Redwood87 1) Water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second, 550 during the third and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m. 2) A firm starts work with 110 employees for the 1st week. the number of the employees rises by 6% for the first week. how many persons will be employed in the 20th week if the present rate of expansion continues. 3) a contractor hires out machinery. in the first year of hiring out one piece of equipment the profit is$6000, but this diminishes by 5% in successive years. show that the annual profits form a geometric progression and find the total of all the profits for the first 5 years.

The first is an arithmetic sequence. Evaluate however many terms are needed in your arithmetic series to get a capacity (sum) of 1 296 000 L.
• November 12th 2012, 07:52 AM
cac2008
Re: Series :) enjoy
Second is a geometric series: determine Sum to however many people are present
Third is also a geometric series. Demonstrate equal ratios between successive terms and compute the sum.
• November 12th 2012, 07:53 AM
Redwood87
Re: Series :) enjoy
how do you evaluate how many terms though?? thats the bit im stuck on..
• November 12th 2012, 09:13 AM
cac2008
Re: Series :) enjoy
1) Total volume = 16x9x9 = 1296 cubic metres = 1296000 litres

$a_{1}=150$

$a_{2}=350$

$a_{3}=350$

Common Difference = $a_{3}-a_{2}=a_{2}-a_{1}=200$

$a_{n}=a_{1}+(n-1)d$ ==> $a_{n}=150+200(n-1)$

Sum to n-terms: $S_{n}=\frac{n}{2}(2a+(n-1)d)$

$\uptherefore 2592000=n(300+200n-200)$

$\frac{2592000}{n}-200n=100$

$\frac{2592000-200n^{2} }{n}=100$

$2592000-200n^{2}=100n$

$-200n^{2}-100n+2592000=0$

$n \approx 113.59,-114.09$

As n must be positive, n=113.59 or 114 (if rounding up)

For 2, use the same process as above, simply replacing the common difference, d, with the common ratio, r such that $r=\frac{a_{n+1}}{a_{n}}$ and repeat, and in 3 rearrange the equation for $S_{n}$ to obtain n, and solve.