The fifth term of an arithmetic progression is 28 and the tenth term in 58.
The sum of all the terms in this progression is 444. How many terms are there?
In the previous question the answer shows that the first term is 4 and the common difference is 6. I also do have the answers available if needed.
Please help me! I don't know what to do to answer this. If you could just set me on the right track I could try to solve it myself.

- anonymous

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- Nnesha

to find common difference \[\huge\rm \frac{ a_{10} -a_5 }{ 10-5 }\]
difference between both terms and the number of terms

- Nnesha

a_10 = 58 and a_5 =28 so subtract \[\huge\rm \frac{ 58-28 }{ 10-5 }\]

- anonymous

Ok, so you get the common difference of 6.

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## More answers

- Nnesha

yes right now we to find number of the terms use the sum formula for arithmetic\[S_n =\frac{ n(a_1 +a_n) }{ 2 }\]

- Nnesha

and i just noticed we don't have the first term hm

- anonymous

yes we do- its 4

- Nnesha

we can use the formula to find term \[\huge\rm a_n =(r)^{n-k}\]

- Nnesha

opps sorry i was looking at the wrong page thats for geometric sequence

- Nnesha

alright so to find 1st term \[\huge\rm a_5 = a_1 +(n-1)d\] d= 6
and we are using 5th term to find first one so n would be 5
\[\huge\rm 28=a_1+(5-1)6\] solve for a_1

- Nnesha

btw you can use 10th term doesn't matter u will get the same answer :=)

- Nnesha

oh!! you already found it yes it's 4

- Nnesha

can i see the answer for last part
little confused on that part

- anonymous

The last part? You mean the first question that I didn't include?

- Nnesha

`How many terms are there? `
i assumed previous question was the same like this one bec we got same value for d and a_1

- anonymous

The entire question is:
7. The fifth term of an arithmetic progression is 28 and the tenth term is 58
(i) Find the first term and the common difference.
(ii) The sum of all the terms in this progression is 444. How many terms are there?
Answers to 7. (i) common difference=6 first term=4

- anonymous

So I got help yesterday for the first one and now I m stuck on how to find the answers for (ii)

- Nnesha

ohh i see..

- Nnesha

alright so we should use \[\huge\rm s_n =\frac{ n(a_1-a_n )}{ 2 }\] a_1 is 4
replace a_n with the a_1 equation which is a_1 = 4+(n-1)6
then solve for n
s_n=144

- Nnesha

444**

- anonymous

Ok I will try that quickly

- Nnesha

try to work on it
ive to go to eat something BRB

- anonymous

sure

- anonymous

Ok I also actually have to be away for a little while. Will come back to this as soon as I can.

- Nnesha

alright tag me when u r ready :=)

- Nnesha

ie found this http://math.stackexchange.com/questions/709276/arithmetic-sequence-find-term-given-sum-of-terms-a1-and-d you should get quadratic equation hm

- anonymous

I am back now @Nnesha , sorry for taking up so much of your time!

- anonymous

Anyway the quadtratic I ended up with was \[-7n^{2}+2n +888\] which looks a bit odd. Did I go wrong somewhere?

- anonymous

Trying to solve it now

- Nnesha

888 ?O_*

- anonymous

Yeah that didn't work

- anonymous

I can send a picture of my working out

- Nnesha

ohh it supposed to be a_1 `+` a_n

- Nnesha

sorry about taht ..

- anonymous

What do you mean?

- Nnesha

alright so we should use \[\huge\rm444 =\frac{ n(4+(4+(n-1)6 )}{ 2 }\] a_1 is 4
replace a_n with the a_1 equation which is a_1 = 4+(n-1)6

- Nnesha

\(\color{blue}{\text{Originally Posted by}}\) @Nnesha
alright so we should use \[\huge\rm s_n =\frac{ n(a_1-a_n )}{ 2 }\] a_1 is 4
replace a_n with the a_1 equation which is a_1 = 4+(n-1)6
then solve for n
s_n=144
\(\color{blue}{\text{End of Quote}}\)
there supposed to be plus sign

- anonymous

Oooooh ok let me try that quickly

- Nnesha

i'll try it let's see what we get

- anonymous

Now the quadratic I get is \[n ^{2} + \frac{ 1 }{ 3 } -148=0\]

- Nnesha

1/3 ? how did you get that ? o.O

- anonymous

Oh dear. Umm I had -6n^2 -2n +888 and I divided everything by -6

- Nnesha

ohh so 1/3n*

- anonymous

Oh yes, sorry

- Nnesha

\[n ^{2} + \frac{ 1 }{ 3 }n -148=0\] can you solve for n ?

- anonymous

Yup doing that now-almost done

- Nnesha

alright let me know what you get

- anonymous

Ok done- the positive value I got was 12.00333333. Do I round that down to 12?

- Nnesha

looks right \[ 444 =\frac{ n(8+6n-6) }{ 2 } ~~~~~= \frac{ n(6n+2) }{ 2 }\] i did it differently but got same answer so i guess 12 is right :=)

- Nnesha

good job! :=)

- anonymous

Thank you! Again I am sorry for taking up so much of your time. I really appreciate your help. Hope you have a nice day :)

- Nnesha

np & you too!

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