Arithmetic series

**parad80** · May 22nd 2012, 11:42 AM

Hi all, just joined and introduced myself in the lobby.

I have two questions which I have not been able to do the workings out for. I understand the formula to calculate arithmetic series, but I am stumped on these - hope you can help.

1. The sum of an arithmetic series is 270. The common difference is 1 and the first term is 4. Calculate the number of terms in the series.

2. The first term of an arithmetic series is 16. The 30th term is 100. Calculate S30.

Can anybody show me the process they would go through?

Thanks

Parad80.

**skeeter** · May 22nd 2012, 12:11 PM

sum of an arithmetic series ...

$S_n = a_1 + (n-1)d$

you know $S_n$ , $d$ , and $a_1$ ... so, what's the problem with finding $n$ ?

**Soroban** · May 22nd 2012, 05:46 PM

Hello, parad80!

You say you understand the formulas.
I don't understand your difficulty.

1. The sum of an arithmetic series is 270.
The common difference is 1 and the first term is 4.
Calculate the number of terms in the series.

The sum is the first n terms of an arithmetic series: . $S_n \:=\:\frac{n}{2}\big[2a_1 + (n-1)d\big]$

We are given: . $S_n = 270,\;a_1=4,\;d = 1$

We have: . $\frac{n}{2}\big[2(4) + (n-1)1\big] \:=\:270$

$\text{Solve for (positive) }n.$

2. The first term of an arithmetic series is 16.
The 30th term is 100.
Calculate S30

$\text{The }n{th}\text{ term is: }\:a_n \:=\:a_1 + (n-1)d$

We are given: . $a_1 = 16,\;n=30,\;a_{30} = 100$

We have: . $16 + 29d \:=\:100 \;\hdots\;\text{Solve for }d.$

$\text{Then determine }S_{30}$

**skeeter** · May 23rd 2012, 03:29 AM

Originally Posted by skeeter

sum of an arithmetic series ...

$S_n = a_1 + (n-1)d$ ... nope, thinking sum, putting down nth term. no more drinking and deriving.

mea culpa.

**parad80** · May 27th 2012, 10:12 PM

Thanks for your responses:

Don't know how you write your formula's in that font, but I understand it up to this point:

1.
n/2(8+(n-1)1),

n/2(8+(n-1)) = 270

Then what is the next step?

Thanks for your patience!

PD

**linalg123** · May 27th 2012, 11:13 PM

expand out the brackets and use the quadratic formula.
$\frac{n}{2}(8+(n-1)) = 270$
$n(8+(n-1)) = 540$
$8n+ n(n-1) = 540$
$n^2 + 7n - 540 = 0$
now use the quadratic formula

remember the quadratic formula will give you two solutions, only the positive one is a feasible answer.

**parad80** · May 28th 2012, 12:01 PM

thanks Iinalg123

understand up to 7n+n2=540

couldn't you then divide by 7 on both sides?

With n2 + 7n - 540 is there an easy way to work the quadratic formula out?

Can someone please tell me how to write the formula's in the fonts above - thanks.

PD

**linalg123** · May 28th 2012, 11:11 PM

Lets say you have a quadratic $ax^2 + bx + c = 0$

the quadratic formula says the solutions are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

If you use this formula for your quadratic you will get your solution for x.

**parad80** · May 29th 2012, 11:37 AM

ok, so I have the following:

-7 + SQRT 7 squared - 4(1)(540)
divided by
2(1)

Then:
-7 + SQRT 49 -2160
divided by
2

Then:
-7 + SQRT - 2111
divided by
2

Then:
-7 + 45.95
divided by
2

Then:
38.95
divided by
2

Then:
19.475, which rounded up is 20, which is the text book answer - thank you.

**linalg123** · May 29th 2012, 06:58 PM

Close, however you didn't convert it to the form $ax^2+bx+c=0$ , you used $ax^2 + bx = c$
So, if we convert it first we have $n^2 + 7n -540 = 0$ (note the negative sign for c)

So, $n = \frac{-7 \pm \sqrt{7^2 - 4(1)(-540)}}{2(1)}$

i.e $n = \frac{-7 \pm \sqrt{49 +2160)}}{2}$ (note the sign chance as negative * negative is positive.)

$n = \frac{-7 \pm \sqrt{2209}}{2}$

$n = \frac{-7 \pm 47}{2}$

now as we only considering the positive answer $n = \frac{40}{2} = 20$

**parad80** · May 29th 2012, 09:59 PM

With regard to the second question:

The first term of an arithmetic series is 16. The 30th term is 100. Calculate S30

I have the following formula:

$S_n \:=\:\frac{n}{2}\big[2a + (n-1)d\big]$

so

$S_1 \:=\:\frac{1}{2}\big[2(16) + (1-1)d\big]$ $\:=\:\frac{1}{2}\big[32 + d\big]$ $\:=\:\big[16 + \frac{1}{2}d\big]$

and

$S_3_0 \:=\:\frac{30}{2}\big[2(100) + (30-1)d\big]$ $\:=\:\frac{30}{2}\big[200 + 29d\big]$ $\:=\:\15\big[200 + 29d\big]$ $\:=\:\big[3000 + 435d\big]$

Where from here? Can't help feeling I am going about this in the wrong way.

PD

**parad80** · May 29th 2012, 10:02 PM

Close, however you didn't convert it to the form. If we convert it first we have (note the negative sign for c)

i.e (note the sign chance as negative * negative is positive.)

Got it - thanks.

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