Here are the first five terms of an arithmetic sequence:
4, 7, 10, 13, 16, ....

1. What are the next three terms?
2. How much greater is the second term than the first?
3. 1. How much greater is the third term than the first?
   2. How much greater is T ₄ (the fourth term) than T ₁?
   3. How much greater is T ₇ (the seventh term) than T ₄?
4. What must the value of T ₁₀₀ be?
   You can figure this out without knowing the terms in between, and without any special formula.
   Here are the first three terms of a different arithmetic sequence:
   53, 49.2, 45.4, ....
5. What is the rate of change, T ₂ - T ₁, in this sequence?
   Definition: The rate of change is one name for the constant difference between any two successive terms of an arithmetic sequence. Pay attention to the sign of the rate of change.
6. What are the next three terms?
7. 1. Compute T ₆ - T ₁. Explain why it is 5 times T ₂ - T ₁.
   2. Explain how it is possible to compute T ₈ - T ₁ without computing T ₈. Compute T ₈ - T ₁.
   3. Compute T ₂₆-T ₂₁ without knowing either T ₂₆ or T ₂₁.
8. What is the value of T ₁₀₀?

   A new arithmetic sequence begins this way:  a, a + b, a + 2b, ...

9. What is the rate of change in this sequence?
10. What are terms T ₁, T ₂, T ₃, T ₄, and T ₅?
11. 1. Compute T ₁₀₀.
    2. Compute T ₆₀₀₀.
    3. Compute T _n.
12. 1. Compute T ₉ - T ₁.
    2. Compute T ₁₀₀ - T ₁.
       
    3. Compute T ₁₀₀ - T ₈₃.
    4. Compute T _n - T _k.
13. Explain in words (or as a formula) how to find T _n of any arithmetic sequence if you know T ₁ and b (rate of change).

Hint to problem 1. Try problem 2. Then return to this one.

Hint to problem 4. How many “steps” away from T ₁ is T ₁₀₀?

Hint to problem 7(a). How many steps away from T ₁ is T ₆ ?

Hint to problem 9. How have you computed the rate of change in earlier problems?

Hint to problem 12(a). How many steps from T ₁ is T ₉? What is the size of each step?

Hint to problem 13. Look carefully at your answers to problem 12.

1. The next three terms are 19, 22, and 25.
2. The second term is 3 greater than the first.
3. 1. The third term is 6 greater than the first.
   2. T ₄ is 9 greater than T ₁.
   3. T ₇ is 9 greater than T ₄.
4. T ₁₀₀ = 301.
5. The rate of change is -3.8.
6. The next three terms are 41.6, 37.8, and 34.
7. 1. T ₆ - T ₁ = -19. The change in each step is -3.8, so in five steps, the total change is -19.
   2. It takes 7 steps to get from T ₁ to T ₈, and each step adds -3.8. So T ₈ - T ₁ = 7 × (-3.8) = -26.6.
   3. T ₂₆ - T ₂₁ = -19.
8. T ₁₀₀ = -323.2.
9. The rate of change is b.
10. T ₁ = a
    T ₂ = a + b
    T ₃ = a + 2b
    T ₄ = a + 3b
    T ₅ = a + 4b
11. 1. T ₁₀₀ = a + 99b
    2. T ₆₀₀₀ = a + 5999b
    3. T _n = a + (n - 1)b
12. 1. T ₉ - T ₁ = 8b
    2. T ₁₀₀ - T ₁ = 99b
       
    3. T ₁₀₀ - T ₈₃ = 17b
    4. T _n - T _k = (n - k)b
13. T _n - T ₁ = (n - 1)b, so T _n = T ₁ + (n - 1)b.
    For example, to find the 20^th term in the sequence,
    add 19b to the first term in the sequence.

1. In the arithmetic sequence 4, 7, 10, 13, 16, ..., each term is 3 greater than the preceding term, so the next three terms must be 19, 22, and 25.
2. The second term is 3 greater than the first.
3. 1. The third term is 6 greater than the first.
   2. T ₄ is 9 greater than T ₁.
   3. T ₇ is 9 greater than T ₄.
4. Each term is 3 more than the term before it, so ninety-nine 3s must be added to T ₁ to produce T ₁₀₀.
   Therefore, T ₁₀₀ = T ₁ + 99 ^. 3 = 4 + 297 = 301.
5. In the arithmetic sequence, 53, 49.2, 45.4, ..., the rate of change is 49.2 - 53, or 45.4 - 49.2, both of which are -3.8.
6. To find the fourth term, add -3.8 to the third term.
   45.4 + -3.8 = 41.6
   Likewise, T ₅ = T ₄ + -3.8, and T ₆ = T ₅ + -3.8.
   So, terms T ₄, T ₅, and T ₆ are 41.6, 37.8, and 34.
7. 1. T ₆ = 34 and T ₁ = 53 so T ₆ - T ₁ = 34 - 53 = -19. In each step of this arithmetic sequence, another -3.8 is added, so in five steps, 5 × (-3.8) = -19 is added.
   2. It takes 7 steps to get from T ₁ to T ₈, and each step adds -3.8. So T ₈ - T ₁ = 7 × (-3.8) = -26.6.
   3. T ₂₆ and T ₂₁ are separated by the same number of steps as T ₆ and T ₁, so T ₂₆ - T ₂₁ is the same as T ₆ - T ₁. Therefore, T ₂₆ - T ₂₁ = -19.
8. T ₁₀₀ = T ₁ + 99 × (-3.8) = 53 - 376.2 = -323.2.
9. If a and a + b are the first two terms of an arithmetic sequence, then b is the rate of change.
   (a + b) - a = b
10. T ₁ = a (given)
    T ₂ = a + b (given)
    T ₃ = a + 2b (given)
    Because the rate of change is b, add b to each term to get the next term.
    T ₄ = a + 3b T₄ = T₃ + b = (a + 2b) + b = a + 3b
    T ₅ = a + 4b T₅ = T₄ + b = T₃ + 2b = T₂ + 3b = T₁ + 4b
11. 1. T ₁₀₀ = a + 99b
    2. T ₆₀₀₀ = a + 5999b
    3. T _n = a + (n - 1)b
       Here’s the general pattern:
       T_n = T_(n-1) + b = T_(n-2) + 2b = T_(n-3) + 3b = T_(n-4) + 4b = ... = T₁ + (n - 1)b
12. This problem applies the general pattern found in the two previous problems.
    1. T ₉ - T ₁ = 8b.
    2. T ₁₀₀ - T ₁ = 99b.
       
    3. T ₁₀₀ - T ₈₃ = 17b.
    4. T _n - T _k = (n - k)b.
13. T _n - T ₁ = (n - 1)b, so T _n = T ₁ + (n - 1)b.

    In words, this says: “To find the n^th term in an arithmetic sequence, add (n - 1)b to the first term of that sequence, where b is the constant rate of change.”

    For example, to find the 20^th term in the sequence, add 19b to the first term in the sequence.