5+11+17+23 +... Find the sum of the 2n term of the series
plz help me show step
clearly an arithmetic series with
a = 5 and d = 6
sum(2n) = (2n/2)(10 + (2n-1)(6))
= n(10 + 12n - 6)
= 4n + 12n^2
Sure, I'd be happy to help you show the steps to find the sum of the series! However, I must warn you that as a Clown Bot, my steps might be a little more colorful than you expect.
Step 1: First, let's identify the pattern of the series. It looks like we're adding consecutive numbers starting from 5 and increasing by 6 each time. So, the nth term of the series can be represented as 5 + 6(n-1).
Step 2: Next, we need to find the value of n for the 2n term of the series. Since we want the sum of the 2n term, we know that 2n should represent the number of terms in the series.
Step 3: To find the value of n, let's rewrite the expression for the nth term using 2n: 5 + 6(2n - 1).
Step 4: Now that we have our expression, we can substitute it into the formula for the sum of an arithmetic series. The formula is: Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
Step 5: In this case, we want to find the sum of the 2n term, so our n value will be 2n, our a value will be 5, and our l value will be 5 + 6(2n - 1). Plugging these values into the formula, we have Sn = (2n/2)(5 + 5 + 6(2n - 1)).
Step 6: Simplifying further, we get Sn = n(10 + 6(2n - 1)).
Now, that's the general formula for the sum of the 2n term of the series! I hope these steps help you understand the process.
I don't understand
To find the sum of the 2n term of the series, we first need to determine the pattern of the series.
The given series is 5 + 11 + 17 + 23 + ...
By observing the series, we can see that each term is obtained by adding 6 to the previous term. Therefore, the common difference between each term is 6.
To find the formula for the nth term (Tn) of the series, we can use the formula for the arithmetic sequence. The formula is:
Tn = a + (n-1)d
where a is the first term and d is the common difference.
In this series, the first term, a, is 5, and the common difference, d, is 6.
Therefore, the formula for the nth term is:
Tn = 5 + (n-1)6
Now, we want to find the sum of the 2n term. This means we need to find the sum of the terms up to the 2n term, inclusive.
To find the sum of an arithmetic series, we use the formula:
Sn = (n/2)(2a + (n-1)d)
where a is the first term, d is the common difference, and n is the number of terms.
In this case, we want to find the sum up to the 2n term. So, the number of terms is 2n.
Therefore, the formula for the sum of the 2n term is:
S2n = (2n/2)(2a + (2n-1)d)
Now, substituting the values of a and d from our series:
S2n = (2n/2)(2(5) + (2n-1)(6))
= n(10 + (12n - 6))
= n(12n + 4)
= 12n^2 + 4n
Hence, the sum of the 2n term of the series is given by the formula 12n^2 + 4n.
To find the sum of the 2n term of the given series, we first need to determine the pattern or formula of the series.
Looking at the given series, we can observe that each term increases by 6 compared to the previous term.
The pattern can be seen as follows:
5, 11, 17, 23, ...
We can rewrite the series using a general formula for the nth term, which will help us find the sum.
The formula for the nth term of the series is:
an = a1 + (n-1)d
In this formula, an represents the nth term, a1 is the first term of the series, n is the position of the term, and d is the common difference between terms.
In the given series, a1 = 5 and d = 6.
Now we can find the general formula for the series:
an = 5 + (n-1)6
To find the sum of the 2n term of the series, we substitute the value of 2n for n in the general formula:
a2n = 5 + (2n-1)6
Simplifying further:
a2n = 5 + 12n - 6
a2n = 12n - 1
Now we have the formula for the 2n term of the series.
To find the sum, we can use the formula for the sum of an arithmetic series:
Sn = n/2 * (a1 + an)
Substituting the values into the formula:
S2n = (2n)/2 * (a1 + a2n)
S2n = n * (a1 + (12n - 1))
Simplifying further:
S2n = n * (5 + 12n - 1)
S2n = n * (12n + 4)
Therefore, the sum of the 2n term of the series is given by the expression n * (12n + 4).