What is the sum of the arithmetic sequence 153, 139, 125, …, 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: 153, 139, 125, ... with a first term (${�}_1$) of 153 and a common difference ($�$) of -14 (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:

${�}_{22}=\frac{22}{2}[2\cdot 153+(22-1)\cdot (-14)]$

Let's break down the calculation step by step:

Calculate $2{�}_1$: $2\cdot 153=306$

Calculate $�-1$: $22-1=21$

Calculate $2{�}_1+(�-1)�$: $306+21\cdot (-14)=306-294=12$

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

Finally, calculate ${�}_{22}$ by multiplying $\frac{�}{2}$ with $(2{�}_1+(�-1)�)$: ${�}_{22}=11\cdot 12$

Now, calculate ${�}_{22}$:

${�}_{22}=11\cdot 12=132$

So, the sum of the arithmetic sequence 153, 139, 125, ... with 22 terms is 132.