Content review: Sequence Connections

✓ Correct: Correct: a₁ = 7, the value at n = 1.

✗ Incorrect: The first term is the value at n = 1.

✗ If 4: That is the value at n = 0, one step before the sequence starts. The first term is the value at n = 1.

✗ If 10: That is the second term. The first term is the value at n = 1.

✓ Correct: Correct: each term increases by 3, so d = 3.

✗ Incorrect: Subtract a term from the next term to find d.

✗ If -3: Check the direction: the terms increase, so d is positive.

✓ Correct: Correct: a₁ = 7; aₙ = aₙ₋₁ + 3, n ≥ 2.

✗ Incorrect: The recursive rule adds the common difference to the previous term.

✗ If [4, any]: The recursive rule must start at the first term a₁ = 7, not at the value at zero. (Incorrect recursive starting term)

✗ If [any, -3]: Check the sign of the change: are the terms going up or down? (Sign error)

✗ If [3, 7]: The starting term and the change are swapped. (Confused slope and intercept)

✗ If [7, any]: The starting term is right. The change is the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, 3]: The change is right. The recursive rule starts at the first term, a₁ = 7. (Incorrect recursive starting term)

✓ Correct: Correct: aₙ = 7 + 3(n − 1).

✗ Incorrect: There are n − 1 steps from term 1 to term n.

✗ If [4, any]: With (n − 1), start from the first term a₁, not the value at zero. (Confused first term with value at zero)

✗ If [any, -3]: Check the sign of the common difference. (Sign error)

✗ If [3, 7]: The first term and the common difference are swapped. (Confused slope and intercept)

✗ If [7, any]: The first term is right. Multiply (n − 1) by the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, 3]: The common difference is right. With (n − 1), start from the first term, a₁ = 7. (Incorrect sequence indexing)

✓ Correct: Correct: f(0) = 7 − 3 = 4. It is the initial value of f(n) = 3n + 4, not a sequence term.

✗ Incorrect: Step back once: f(0) = a₁ − d.

✗ If 7: That is the first term, f(1). Step back one more step to n = 0.

✗ If 10: You stepped forward. To step back, undo one change: f(0) = a₁ − d.

✓ Correct: Connected! The function rule gives f(1) = 10, but the first term is 7. Its constant should be 4, the value at zero.

✗ Incorrect: Test each rule at term 1. Which one fails to produce the first listed term? Check the recursive starting value too.

✓ Correct: Correct: a₁ = 18, the value at n = 1.

✗ Incorrect: The first term is the value at n = 1.

✗ If 22: That is the value at n = 0, one step before the sequence starts. The first term is the value at n = 1.

✗ If 14: That is the second term. The first term is the value at n = 1.

✓ Correct: Correct: each term decreases by 4, so d = −4.

✗ Incorrect: Subtract a term from the next term to find d.

✗ If 4: Check the direction: the terms decrease, so d is negative.

✓ Correct: Correct: a₁ = 18; aₙ = aₙ₋₁ − 4, n ≥ 2.

✗ Incorrect: The recursive rule adds the common difference to the previous term.

✗ If [22, any]: The recursive rule must start at the first term a₁ = 18, not at the value at zero. (Incorrect recursive starting term)

✗ If [any, 4]: Check the sign of the change: are the terms going up or down? (Sign error)

✗ If [-4, 18]: The starting term and the change are swapped. (Confused slope and intercept)

✗ If [18, any]: The starting term is right. The change is the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, -4]: The change is right. The recursive rule starts at the first term, a₁ = 18. (Incorrect recursive starting term)

✓ Correct: Correct: aₙ = 18 + (−4)(n − 1).

✗ Incorrect: There are n − 1 steps from term 1 to term n.

✗ If [22, any]: With (n − 1), start from the first term a₁, not the value at zero. (Confused first term with value at zero)

✗ If [any, 4]: Check the sign of the common difference. (Sign error)

✗ If [-4, 18]: The first term and the common difference are swapped. (Confused slope and intercept)

✗ If [18, any]: The first term is right. Multiply (n − 1) by the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, -4]: The common difference is right. With (n − 1), start from the first term, a₁ = 18. (Incorrect sequence indexing)

✓ Correct: Correct: f(0) = 18 − (−4) = 22. It is the initial value of f(n) = −4n + 22, not a sequence term.

✗ Incorrect: Step back once: f(0) = a₁ − d.

✗ If 18: That is the first term, f(1). Step back one more step to n = 0.

✗ If 14: You stepped forward. To step back, undo one change: f(0) = a₁ − d.

✓ Correct: Connected! The explicit rule gives a₁ = 14, not 18. Use n − 1 because there are n − 1 steps from the first term.

✗ Incorrect: Test each rule at term 1. Which one fails to produce the first listed term? Check the recursive starting value too.

✓ Correct: Correct: a₁ = 12, the value at n = 1.

✗ Incorrect: The first term is the value at n = 1.

✗ If 7: That is the value at n = 0, one step before the sequence starts. The first term is the value at n = 1.

✗ If 17: That is the second term. The first term is the value at n = 1.

✓ Correct: Correct: each term increases by 5, so d = 5.

✗ Incorrect: Subtract a term from the next term to find d.

✗ If -5: Check the direction: the terms increase, so d is positive.

✓ Correct: Correct: a₁ = 12; aₙ = aₙ₋₁ + 5, n ≥ 2.

✗ Incorrect: The recursive rule adds the common difference to the previous term.

✗ If [7, any]: The recursive rule must start at the first term a₁ = 12, not at the value at zero. (Incorrect recursive starting term)

✗ If [any, -5]: Check the sign of the change: are the terms going up or down? (Sign error)

✗ If [5, 12]: The starting term and the change are swapped. (Confused slope and intercept)

✗ If [12, any]: The starting term is right. The change is the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, 5]: The change is right. The recursive rule starts at the first term, a₁ = 12. (Incorrect recursive starting term)

✓ Correct: Correct: aₙ = 12 + 5(n − 1).

✗ Incorrect: There are n − 1 steps from term 1 to term n.

✗ If [7, any]: With (n − 1), start from the first term a₁, not the value at zero. (Confused first term with value at zero)

✗ If [any, -5]: Check the sign of the common difference. (Sign error)

✗ If [5, 12]: The first term and the common difference are swapped. (Confused slope and intercept)

✗ If [12, any]: The first term is right. Multiply (n − 1) by the common difference: subtract a term from the next term. (Incorrect common difference)

✗ If [any, 5]: The common difference is right. With (n − 1), start from the first term, a₁ = 12. (Incorrect sequence indexing)

✓ Correct: Correct: f(0) = 12 − 5 = 7. It is the initial value of f(n) = 5n + 7, not a sequence term.

✗ Incorrect: Step back once: f(0) = a₁ − d.

✗ If 12: That is the first term, f(1). Step back one more step to n = 0.

✗ If 17: You stepped forward. To step back, undo one change: f(0) = a₁ − d.

✓ Correct: Connected! The recursive rule starts at 7, but the given sequence starts at 12. The starting term must be a₁ = 12.

✗ Incorrect: Test each rule at term 1. Which one fails to produce the first listed term? Check the recursive starting value too.

✓ Correct: Connection made! The function rule does not belong: f(1) = 16, but the first term is 10. The matching function is f(n) = 6n + 4. Its initial value is 4; its first sequence term is 10.

✗ Incorrect: Test each rule at term 1. Which one fails to produce the first listed term? Check the recursive starting value too.