Question

What is the sum of the first 30 terms of the arithmetic series 27,32,37,42,47

Answer 1

Answer:

2985

Step-by-step explanation:

The formula for the sum of an arithmetic series is given by the formula:

$S=\frac{k}{2}(a+x_k)$

Where k is the number of terms, a is the first term, and x_k is the last term. In this case, the last term is the 30th term because we are finding the sum of the first 30 terms.

So, we need to find the 30th term. To do so, we can write an explicit formula for our sequence.

The standard form for an arithmetic sequence is given by the formula:

$x_n=a+d(n-1)$

Where a is the initial term and d is the common difference.

From our sequence, we can see that the initial term a is 27. The common difference is 5 because each term is 5 greater than the previous one.

So, our equation is:

$x_n=27+5(n-1)$

So, to find the 30th term, substitute 30 for n:

$x_{30}=27+5(30-1)$

Subtract:

$x_{30}=27+5(29)$

Multiply:

$x_{30}=27+145$

Add:

$x_{30}=172$

So, the 30th term is 172.

Now, substitute this into our original sum formula:

$S=\frac{k}{2}(a+x_k)$

Substitute 30 for k (the amount of terms), 27 for a (the first term), and 172 for x_k (the 30th and last term). So:

$S=\frac{30}{2}(27+172)$

Reduce and add:

$S=15(199)$

Multiply:

$S=2985$

So, the sum of the first 30 terms is 2985.