Help With Question (series) (1 Viewer)

[sando]

Hi this is one of my h/w questions and i think i know the process of it but i just cant seem to get the write answer. The question is:

1. Find the first vale of n for which the sum of the series 20 + 4 + 4/5 + ... is greater than 24.85


Any help would be appreciated

 

[insert-username]

1. Find the first vale of n for which the sum of the series 20 + 4 + 4/5 + ... is greater than 24.85

Simple solution: It's the 4th term (n=4).

a = 20 (first term)
r = 1/5 (common ratio)

The first 3 terms (given) add up to 24.80. Calculating the fourth term gives us 4/25, so the sum of 4 terms gives 24.96, which is greater than 24.85.


Harder solution: (this solution involves logarithms)

a = 20 (first term)
r = 1/5 (common ratio)
n is an integer

Sum of n terms of a geometric series (|r| < 1) = [a(1-rⁿ)] ÷ (1-r)

= [100(1-1/5ⁿ)] ÷ 4 (subbing in values of a and r)

= 25(1-1/5ⁿ) (dividing by 4)


We want to find the term which is greater than 24.85, therefore:

24.85 < 25(1-1/5ⁿ)

0.994 < (1-1/5ⁿ) (dividing by 25)

1/5ⁿ < 0.006 (making 1/5ⁿ the subject)


Rewrite as a logarithm:

log_1/5 0.006 < n


Change the base to base 10:

(log₁₀ 0.006) ÷ (log₁₀ 1/5) < n (using change of base formula)

3.1787 < n

Therefore n (which must be an integer) = 4.


I_F

 

[sando]

yeah i just realised how to work it out using the first method. Thanks

But i havnt even learnt log yet. Someone told me i was suppossed to learn it in yr 10.


Am i screwed???

 

[m0ofin]

.

 

[Riviet]

Whooo! Logs and exponentials are fun!

Wait till you get to differentiating them, and i say that in a positive way.

 

[sando]

Whooo! Logs and exponentials are fun! (no sarcasm intended) But Locus sucks big time ><!!

Agree. Locus really really sucks.

 

[Riviet]

Hence evaluate - the two worst words in the English language

Tip for any "hence [insert verb here]" type question in maths: it will always involve the previous part in helping you evaluate whatever it is. Usuaully the question will ask you to use part whatever to do the next bit, so when you answer it, you need to say "from part a" or wherever you used something found in a previous part.

 

[insert-username]

Technically, we haven't done logs yet either. I've just done some extension work along the way (the sad case that I am). It's an option topic in Year 10, but most people cover it in Year 12, so don't worry too much.


I_F

 

[Riviet]

I've just done some extension work along the way (the sad case that I am)

Good on ya for taking the initiative to work ahead, that's what i'm doing at the moment with holidays.

 

[insert-username]

Hehe... good on you too. Right now I'm catching up on the work I couldn't be bothered doing in the last week of term... But give me a few weeks and I'll start pushing ahead again.


I_F