# Thread: Simple Arithmetic Series Question

1. ## Simple Arithmetic Series Question

In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th term.

How do I do this?

2. Originally Posted by Viral
In an arithmetic series the 20th term is 14 and the 40th term is -6. Find the 10th term.

How do I do this?
$U_n = a+(n-1)d$

Where

• $U_n$ is the nth term
• $a$ is the first term
• $n$ is the number of terms
• $d$ is the common difference.

$U_{20} = a + (20-1)d = 14$ (eq1)

$U_{40} = a + (40-1)d = -6$ (eq2)

Solve simultaneously. It would be easiest to eliminate a first

3. Yeah, I've done that, and got $d = 1 \therefore a = -5$.

However, when I do:

$U_{10}=-5+9d=14=v$
$v=-5+9=4$

But the answer is supposed to be 24 >< .

4. Originally Posted by Viral
Yeah, I've done that, and got $d = 1 \therefore a = -5$.

However, when I do:

$U_{10}=-5+9d=14=v$
$v=-5+9=4$

But the answer is supposed to be 24 >< .
$U_{20} > U_{40}$ which means $d<0$

I took equation 2 from equation 1:

$(a+19d) - (a+39d) = 14 - (-6)$

$-20d = 20 \: \therefore \: d = -1$

$a+19(-1) = 14 \: \therefore \: a = 33$

$U_{10} = a + 9d = 33-9 = 24$

5. I'm now told: The first three terms of an arithmetic series are 5x, 20 and 3x. Find the values of x and hence the values of the three terms.

What are the steps to do this?

6. Originally Posted by Viral
I'm now told: The first three terms of an arithmetic series are 5x, 20 and 3x. Find the values of x and hence the values of the three terms.

What are the steps to do this?
By definition: $a = 5x$

$U_2 = 20 = a+d$

$U_3 = 3x = a+2d$

You can then solve U_2 and U_3 simultaneously. Eliminate d since we don't need to find it

I get $x=5$ and so $a = 25$, $U_2 = 20$ and $U_3 = 15$

7. Ok, this one is worded really weirdly >< . For this can you just tell me what it's asking, and I'll go from there.

For which values of x would the expression $-8, x^{2} and 17x$ form the first three terms of an arithmetic series.

8. Originally Posted by Viral
Ok, this one is worded really weirdly >< . For this can you just tell me what it's asking, and I'll go from there.

For which values of x would the expression $-8, x^{2} and 17x$ form the first three terms of an arithmetic series.
Do you get what I did above with the first three terms? In this case set the numbers you're given here

$U_1 = a = -8$

$U_2 = a+d = -8+d = x^2$

$U_3 = a+2d = -8+2d = 17x$