Question

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.

Answer 1

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

Answer 2

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

Answer 3

Ok, so you get the common difference of 6.

Answer 4

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

Answer 5

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

Answer 6

yes we do- its 4

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

`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

Answer 15

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

Answer 16

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

Answer 17

ohh i see..

Answer 18

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

Answer 19

444**

Answer 20

Ok I will try that quickly

Answer 21

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

Answer 22

sure

Answer 23

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

Answer 24

alright tag me when u r ready :=)

Answer 25

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

Answer 26

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

Answer 27

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

Answer 28

Trying to solve it now

Answer 29

888 ?O_*

Answer 30

Yeah that didn't work

Answer 31

I can send a picture of my working out

Answer 32

ohh it supposed to be a_1 `+` a_n

Answer 33

sorry about taht ..

Answer 34

What do you mean?

Answer 35

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

Answer 36

\(\color{blue}{\text{Originally Posted by}}\) @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

Answer 37

Oooooh ok let me try that quickly

Answer 38

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

Answer 39

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

Answer 40

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

Answer 41

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

Answer 42

ohh so 1/3n*

Answer 43

Oh yes, sorry

Answer 44

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

Answer 45

Yup doing that now-almost done

Answer 46

alright let me know what you get

Answer 47

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

Answer 48

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 :=)

Answer 49

good job! :=)

Answer 50

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 :)

Answer 51

np & you too!