How do you write a rule for the nth term of the arithmetic sequence given a20 = 240 and a15 = 170?

To find the rule for the nth term of the given arithmetic sequence, we start by identifying the information provided:

• A₂₀ = 240
• A₁₅ = 170

In an arithmetic sequence, the nth term can be expressed with the formula:

A_n = A₁ + (n – 1) * d

where A₁ is the first term and d is the common difference between the terms.

From the given terms A₂₀ and A₁₅, we can express them using the nth term formula:

• A₂₀ = A₁ + 19d = 240
• A₁₅ = A₁ + 14d = 170

Now, we have a system of equations:

A1 + 19d = 240

A1 + 14d = 170

By subtracting the second equation from the first, we can eliminate A₁:

(A1 + 19d) - (A1 + 14d) = 240 - 170

5d = 70

Solving for d gives:

d = 14

Now that we have the common difference, we can substitute d back into one of the equations to find A₁. Using the second equation:

A1 + 14 * 14 = 170

A1 + 196 = 170

A1 = 170 - 196

A1 = -26

Now that we have both A₁ and d, we can write the rule for the nth term:

A_n = -26 + (n – 1) * 14

Therefore, the rule for the nth term of the arithmetic sequence is:

A_n = -26 + 14(n – 1)