Question

Given the sequence 2,7,12,..., write an explicit
formula for the nth term, assuming a1=2.
2 Given the sequence 2,7,12,..., find the sum of
the first 30 terms
​

Answer 1

Answer:

Part 1) $a_n=2+5(n-1)$

Part 2) The sum of  the first 30 terms is 2,235

Step-by-step explanation:

Part 1)  write an explicit  formula for the nth term

we know that

In an Arithmetic Sequence the difference between one term and the next is a constant, and this constant is called the common difference

We can write an Arithmetic Sequence as a rule:

$a_n=a_1+d(n-1)$

where

$a_n$ is the nth term

$a_1$ is the first term

n is the number of terms

d is the common difference

we have the sequence

$2,7,12,...$

we have

$a_1=2$, $a_2=7$, $a_3=12$

Find the common difference d

$a_2-a_1=7-2=5$

$a_3-a_2=12-7=5$

The common difference  is d=5

substitute in the formula

$a_n=2+5(n-1)$

Part 2) Find the sum of  the first 30 terms

we know that

The formula to calculate the sum of an arithmetic sequence is

$S=\frac{n}{2}(2a_1+(n-1)d)$

where

$a_1$ is the first term

n is the number of terms

d is the common difference

we have

$a_1=2$

$d=5$

$n=30$

substitute

$S=\frac{30}{2}(2(2)+(30-1)5)$

$S=2,235$