What is the sum of the arithmetic sequence 8, 14, 20 …, if there are 22 terms? Explain step by step in detail.

To find the sum of an arithmetic sequence, you can use the formula:

${�}_{�}=\frac{�}{2}[2{�}_1+(�-1)�]$

Where:

• ${�}_{�}$ is the sum of the first $�$ terms.
• $�$ is the number of terms.
• ${�}_1$ is the first term in the sequence.
• $�$ is the common difference between terms.

In your case, you have an arithmetic sequence: 8, 14, 20, ... with a first term (${�}_1$) of 8 and a common difference ($�$) of 6 (the difference between consecutive terms). You want to find the sum (${�}_{�}$) of this sequence with 22 terms ($�=22$).

Now, let's plug these values into the formula:

${�}_{22}=\frac{22}{2}[2\cdot 8+(22-1)\cdot 6]$

Let's break down the calculation step by step:

Calculate $2{�}_1$: $2\cdot 8=16$

Calculate $�-1$: $22-1=21$

Calculate $2{�}_1+(�-1)�$: $16+21\cdot 6=16+126=142$

Calculate $\frac{�}{2}$: $\frac{22}{2}=11$

Finally, calculate ${�}_{22}$ by multiplying $\frac{�}{2}$ with $(2{�}_1+(�-1)�)$: ${�}_{22}=11\cdot 142$

Now, calculate ${�}_{22}$:

${�}_{22}=11\cdot 142=1562$

So, the sum of the arithmetic sequence 8, 14, 20, ... with 22 terms is 1562.