The sum of the first 20 terms of an arithmetic series is identical to the sum of the first 22 terms. If the common difference is -2, find the first term.
sum of 20 = 10(2a + 19d)
= 10(2a - 38)
= 20a - 380
sum of 22 = 11(2a + 21d)
= 11(2a - 42)
22a - 462
22a - 462 = 20a - 380
2a = 82
a = 41
first term is 41
I bet the the series near zero does this...3, 1,-1, -3
Now use your sum series to find the first term if the twentth term is 3.
Well, it seems like the arithmetic series is playing a game of hide and seek with us. But don't worry, I'll help you find the first term!
Let's call the first term "a". Since we know the common difference is -2, we can write the formula for the nth term as:
an = a + (n - 1)d
where d is the common difference.
So for the first 20 terms, we have:
20a = a + (20 - 1)(-2)
20a = a - 38
Now, let's write the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n - 1)d)
For the sum of the first 20 terms, we have:
20/2(2a + (20 - 1)(-2)) = 22/2(2a + (22 - 1)(-2))
10(2a - 38) = 11(2a - 40)
20a - 380 = 22a - 440
22a - 20a = 440 - 380
2a = 60
a = 30
So the first term of the arithmetic series is 30. I hope my calculation didn't subtract too much fun from your day!
To find the first term of the arithmetic series, we need to set up an equation using the given information.
The sum of an arithmetic series can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Where:
- Sn is the sum of the first n terms of the arithmetic series,
- a is the first term,
- d is the common difference, and
- n is the number of terms.
In this case, we have:
S20 = S22
Using the formula mentioned earlier, we can write the equation as:
(20/2)(2a + (20-1)(-2)) = (22/2)(2a + (22-1)(-2))
Simplifying this equation, we get:
10(2a + 19(-2)) = 11(2a + 21(-2))
Now, we can solve for a:
10(2a - 38) = 11(2a - 42)
Expanding the equation:
20a - 380 = 22a - 462
Rearranging terms:
22a - 20a = 462 - 380
2a = 82
Dividing both sides by 2:
a = 41
Therefore, the first term of the arithmetic series is 41.