Series and Sequences (1 Viewer)

[Aerath]

So lost with this question... =(

The rain drops falling per minute forms the series
1+1+3+9+23+...
with the nth term given by the formula
R_n = 1-2n+2ⁿ

i) Verify that 115 is a term of this series

ii) Find the total amount of raindrops which fall over the river in the first 25 minutes

iii) If the surface area of the river is 250m^2 find the average rainfall per cm^2 over the first 25 minutes

 

[lyounamu]

So lost with this question... =(

The rain drops falling per minute forms the series
1+1+3+9+23+...
with the nth term given by the formula
R_n = 1-2n+2ⁿ

i) Verify that 115 is a term of this series

ii) Find the total amount of raindrops which fall over the river in the first 25 minutes

iii) If the surface area of the river is 250m^2 find the average rainfall per cm^2 over the first 25 minutes

i)

1-2n + 2^n = 115 when n=k
115 = 1-2k+2^k
114 = -2k + 2^k
Trial and error method:
Therefore, 1-2n + 2^n = 115 when (n = k) = 7

It's a "verify" question so I think all you need to do is to show that 115 is a term in the series by working out the series itself.

ii)
Since the series is neither geometric nor arithmic, I divided the series and work out the sum separately and add them up.
Let Tn1 = 1 - 2n and Tn2 = 2^n
i.e. Sn = Sn1 + Sn2
Sn1 = 25/(2(-1)+(25-1)-2) = -625
Sn2 = 2(2^25-1)/(2-1) = 67108862

Sn = 67108862 - 625 = 67108237

iii) Since no measurement is given, I will assume that it is mL. 67108237 mL
Divide that by 25000000 = 2.6843294... = 2.69 mL

I divided by 2500000 because 250 m^2 = 2500000 cm^2.

 

[5647382910]

i)

1-2n + 2^n = 115 when n=k
115 = 1-2k+2^k
114 = -2k + 2^k
Trial and error method:
Therefore, 1-2n + 2^n = 115 when (n = k) = 7

It's a "verify" question so I think all you need to do is to show that 115 is a term in the series by working out the series itself.

ii)
Since the series is neither geometric nor arithmic, I divided the series and work out the sum separately and add them up.
Let Tn1 = 1 - 2n and Tn2 = 2^n
i.e. Sn = Sn1 + Sn2
Sn1 = 25/(2(-1)+(25-1)-2) = -625
Sn2 = 2(2^25-1)/(2-1) = 67108862

Sn = 67108862 - 625 = 67108237

iii) Since no measurement is given, I will assume that it is mL. 67108237 mL
Divide that by 25000000 = 2.6843294... = 2.69 mL

I divided by 2500000 because 250 m^2 = 2500000 cm^2.

nice work.... the answer should be 26.8.... u divided by 1 zero too much

 

[lyounamu]

nice work.... the answer should be 26.8.... u divided by 1 zero too much

I wrote it straight on the computer so I was vulnerable to many mistakes anyway.