First term 30 the sum of the first five terms is 300
Well, it seems that you've encountered a math problem! Let me put on my clown nose and help you out with some humor-infused calculations.
If the first term is 30 and the sum of the first five terms is 300, then we can use some clown math to solve this riddle.
Let's call the common difference between the terms as "d", just to keep things fancy. Since we're dealing with arithmetic progression, the sum of the first five terms can be calculated using the formula:
Sum = (n/2) * (2a + (n-1)d),
where "n" is the number of terms, "a" is the first term, and "d" is the common difference.
In this case, we know that the first term is 30 and the sum is 300. Since "n" is 5, we can substitute the given values into the formula:
300 = (5/2) * (2*30 + (5-1)d).
Simplifying this equation, we get:
300 = (5/2) * (60 + 4d),
600 = 300 + 20d,
20d = 300,
d = 15.
So, our final answer is that the common difference, represented by "d", is 15. Now you know that the difference between each term in this progression is no joking matter— it's 15!
To find the common difference, we need to find the formula for the arithmetic sequence. We know the first term is 30, and let's call the common difference "d".
The general formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n-1)d,
where "an" is the nth term, "a1" is the first term, and "d" is the common difference.
We are given that the sum of the first five terms is 300. The formula for the sum of an arithmetic series is:
Sn = (n/2)(a1 + an).
Substituting the values we have:
300 = (5/2)(30 + (30 + 4d)).
Now we can solve for "d".
To find the sum of the first five terms of an arithmetic sequence, you can use the formula:
S = (n/2) * (2a + (n-1)d)
Where:
S is the sum of the terms,
n is the number of terms,
a is the first term, and
d is the common difference between the terms.
In this case, you are given that the sum of the first five terms is 300, and the first term is 30. So, we can substitute the values into the formula:
300 = (5/2) * (2 * 30 + (5-1)d)
Now we can simplify the equation:
300 = (5/2) * (60 + 4d)
600 = 5 * (60 + 4d)
Divide both sides of the equation by 5:
120 = 60 + 4d
Subtract 60 from both sides of the equation:
4d = 60
Divide both sides of the equation by 4:
d = 15
So, the common difference between the terms is 15.
Now that we have the first term (a = 30) and the common difference (d = 15), we can find the remaining terms of the arithmetic sequence if needed.
a = 30
5/2 (2a+4d) = 300
Now solve for d, and list whatever terms you need.