The explicit formula for an arithmetic sequence is aₙ = a₁ + (n − 1)d. A student mistakenly writes aₙ = a₁ + nd instead. What specific error does this introduce?
AThe formula now calculates the sum of the sequence instead of a single term
BThe formula produces a result that is exactly d too large — it adds d one extra time, as if the first term had already had d added to it
CThe formula breaks down only for large values of n
DThe formula confuses the common difference with the first term
The (n − 1) exists because to reach the nth term you add d exactly (n − 1) times — the first term is reached with zero additions of d. Using nd instead adds d one extra time, producing a result that is always d larger than the correct answer. For example, in the sequence 3, 7, 11, 15... with a₁ = 3 and d = 4: the 3rd term should be 11. The correct formula gives 3 + (3−1)×4 = 11; the wrong formula gives 3 + 3×4 = 15 — that's the 4th term, not the 3rd.
Question 2 Multiple Choice
You plot the terms of the arithmetic sequence 5, 8, 11, 14, 17, ... with term number n on the x-axis and term value on the y-axis. What does the graph look like, and what determines its slope?
AA curved (parabolic) line, because the values keep increasing
BPoints falling on a straight line with slope 5, since a₁ = 5
CPoints falling on a straight line with slope 3, since the common difference d = 3
DA scatter plot with no pattern, since sequences are discrete
Arithmetic sequences are discrete linear functions. When plotted, the points fall exactly on a straight line. The slope of that line is the common difference d — in this case, 3. The first term a₁ determines the y-intercept (specifically, the y-intercept of the underlying line is a₁ − d = 5 − 3 = 2). Option B is the classic confusion: a₁ is the value of the first term, not the slope. The rate of change between terms — which is what slope measures — is d.
Question 3 True / False
A decreasing arithmetic sequence (where each term is smaller than the previous) is not truly arithmetic because the common difference is expected to be positive.
TTrue
FFalse
Answer: False
A sequence is arithmetic if the difference between consecutive terms is constant — regardless of sign. A negative common difference (d < 0) produces a decreasing sequence that is perfectly arithmetic. For example, 20, 15, 10, 5, 0, −5, ... has d = −5 and is arithmetic. The formula aₙ = a₁ + (n − 1)d works exactly the same way; d being negative simply means the sequence decreases rather than increases.
Question 4 True / False
An arithmetic sequence is a discrete version of a linear function: the common difference plays the same role as slope.
TTrue
FFalse
Answer: True
The explicit formula aₙ = a₁ + (n − 1)d can be rewritten as aₙ = d·n + (a₁ − d), which is exactly slope-intercept form y = mx + b. The common difference d is the slope (the constant rate of change per step), and (a₁ − d) is the y-intercept. The only difference from a continuous linear function is that n is restricted to positive integers — you only hit discrete points on the line rather than every real x-value.
Question 5 Short Answer
A job starts at a salary of $48,000 and increases by $3,000 every year. Explain why this is an arithmetic sequence and calculate the salary in the 6th year.
Think about your answer, then reveal below.
Model answer: This is arithmetic because the salary increases by a constant amount ($3,000) each year — that constant is the common difference d = 3,000. Using the explicit formula: a₆ = a₁ + (n − 1)d = 48,000 + (6 − 1)(3,000) = 48,000 + 15,000 = $63,000.
The key check for arithmetic sequences is whether the difference between any two consecutive terms is always the same. Here, year 1 to year 2 increases by $3,000, year 2 to year 3 increases by $3,000, and so on — constant difference, arithmetic sequence. The (n − 1) in the formula accounts for the fact that you add d five times to get from year 1 to year 6, not six times.