Find S25 for 1/2 +1+ 3/2 + 2+

Question

anonymous · Answer

is that S  sum or  25 term?

mathstudent55 · Answer

Is this an arithmetic or a geometric series?

anonymous · Answer

just like the last one we did. its S with 25 in the lower right coner and @mathstudent55  its arithmetric

anonymous · Answer

then apply   a+24d

mathstudent55 · Answer

Great.
The formula for the sum of an arithmetic series is:

$\Large S_n = \dfrac{n(a_1 + a_n)}{2} $

We have the first term. We need the 25th term.

anonymous · Answer

is @GIL.ojei way correct too?

anonymous · Answer

yap correct for sum. that was why i asked

mathstudent55 · Answer

Yes, he's correct to find the 25th term.

$\Large a_n = a_1 + (n - 1)d$

anonymous · Answer

okay. so I can use his way then?

mathstudent55 · Answer

First use the formula just above for finding the 25th term.
Then use the formula for $S_n$ to find the sum of the first 25 terms.

anonymous · Answer

follow his way. mine is term and his is sum

anonymous · Answer

i dont think i did the first one right bc i got 318.75 
@mathstudent55 @GIL.ojei  helpp? lol

mathstudent55 · Answer

You are looking for the sum of the first 25 terms of an arithmetic series.
This is the formula you need to use. The general formula is below.

$\Large S_n = \dfrac{n(a_1 + a_n)}{2}$

We replace n with 25 to give you the exact formula you need in your case, to sum the first 25 terms.

$\Large S_{25} = \dfrac{n(a_1 + a_{25})}{2}$

Now you see that you need the first term, which you have, and the 25th term.
For the 25th term, we use the formula below.

This is the general formula you sue to find the nth term of a series.
Since we need the 25th term, we replace n with 25.

$\Large a_{25} = a_1 + (25 - 1)d$

Form the first two given terms, 1/2 and 1, we find d, the common difference: 
d = 1 - 1/2 = 1/2 = 0.5

Now we can find the 25th term:

$\Large a_{25} = 0.5 + (25 - 1)0.5$

$\Large a_{25} = 12.5$

Now we can go back to the sum formula using the first and 25th terms:

$\Large S_{25} = \dfrac{25(0.5 + 12.5)}{2}$

$\Large S_{25} = 162.5$

jim_thompson5910 · Answer

Look at the formulas mathstudent55 wrote out, you will plug $\Large {\color{red}{a_n=a_1+d(n-1)}}$ into the $\Large S_n$ formula to get...

$$\Large S_n = \dfrac{n(a_1 + a_n)}{2}$$

$$\Large S_n = \dfrac{n(a_1 + {\color{red}{a_n}})}{2}$$

$$\Large S_n = \dfrac{n(a_1 + {\color{red}{a_1 + d(n-1)}})}{2}$$

$$\Large S_n = \dfrac{n(2a_1 + d(n-1))}{2}$$

anonymous · Answer

i get it. thank you! sorry for the late responce, i was getting food.