Show work and explain with formulas.

4. if a_6 = 18 and a_13 = 32, find the sum of the 13 terms in an arithmetic sequence.

5. find the sum (see picture).

6. find the first three terms of the arithmetic series: a_1 = 17, a_n = 197, s_n = 2247

Step-by-step explanation:

4.

We have:

These are the terms of the arithmetic sequence.

We know:

Therefore

Substitute a₆ = 18 and a₁₃ = 32:

            divide both sides by 7

           subtract 10 from both sides

The formula of a sum of terms of an arithmetic sequence:

Substitute a₁ = 8, a₁₃ = 32 and n = 13:

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5.

We have

Calculate

Calculate the difference:

It's the arithmetic sequence with first term

and common difference d = 2.

The formula of a sum of terms of an arithmetic sequence:

Substitute n = 7, a₁ = 7 and d = 2:

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6.

We have:

The formula for the n-th term of an arithmetic sequence:

The formula of the sum of terms of an arithmetic sequence:

Substitute:

Convert the first equation:

         subtract 17 from both sides

   Substitute it to the second equation:

            divde both sides by 107

Put the value of n to the equation (n - 1)d = 180:

             divide both sides by 20

Therefore we have the explicit formula for the nth term of an arithmetic sequence:

Put n = 1, n = 2 and n = 3: