Mathematics, four units

How do you find the first term and the common difference from two given terms?

Quick answer

To find the first term and common difference, divide the difference between the two term values by the difference between their position numbers. This division gives the common difference, d. Next, substitute d into the formula for either term, a_n = a_1 + (n - 1)d, and solve for the first term, a_1.

The topic explained

The gap method for finding d

Every arithmetic sequence adds the same fixed value, called the common difference (d), at every step. Because of this regular spacing, any two terms are separated by an exact count of differences equal to the gap between their positions.

To find the first term and the common difference, start with d. Take the difference between the two given term values and divide it by the difference between their index positions: d = (a_k - a_m) / (k - m). For example, if the 5th term is 17 and the 9th term is 33, there are 4 steps between them (9 - 5 = 4). The value gap is 33 - 17 = 16. Dividing 16 by 4 gives d = 4.

Backtracking to the first term

Once you know d, finding the first term (a_1) requires only one substitution. The nth term formula is a_n = a_1 + (n - 1)d. Notice the (n - 1): reaching the nth term from the start always takes n - 1 steps, not n steps.

Using the 5th term from our example, write 17 = a_1 + (5 - 1)(4). This simplifies to 17 = a_1 + 16, which reveals that a_1 = 1. You can verify this using the 9th term: 1 + (9 - 1)(4) = 1 + 32 = 33, which matches perfectly.

A common mistake to avoid

The most frequent mistake in Grade 11 sequence problems is multiplying d by n instead of (n - 1). If you multiply by n, you add an extra step and shift the entire sequence by one place.

Always remember: the first term takes zero steps to reach. The second takes one step, the third takes two, and the nth term takes n - 1 steps. Checking your calculated first term against the second given term catches this error immediately.

Free printable worksheet

Finding a_1 and d in Arithmetic Sequences

Use the given terms to find the common difference (d) and the first term (a_1) for each arithmetic sequence. Write your answers in simplest form.

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Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Finding a_1 and d in Arithmetic SequencesName:

Given a_3 = 11 and a_4 = 15. Find d and a_1.
Given a_2 = 9 and a_6 = 25. Find d and a_1.
Given a_4 = 22 and a_8 = 46. Find d and a_1.
Given a_3 = 20 and a_7 = 8. Find d and a_1.
Given a_5 = -2 and a_11 = -20. Find d and a_1.
Given a_6 = 14.5 and a_10 = 22.5. Find d and a_1.
In an arithmetic sequence, a_7 = 53 and a_13 = 101. Find d, a_1, and the value of a_20.

Common questions

Can the common difference be negative or fractional?

Yes. If the sequence values decrease as the term number increases, d is negative. If the values increase by parts of a whole, d is a fraction or decimal. The formula works identically regardless of sign.

How do I check if a given value belongs to the sequence?

Set the nth term formula a_n = a_1 + (n - 1)d equal to that value and solve for n. If n is a positive integer, the number is in the sequence. If n is fractional or negative, it is not.

Why is the term index always n minus 1?

The sequence starts at the first term without adding any difference. You only add the common difference when advancing to subsequent terms. Therefore, reaching position n requires exactly n - 1 additions.

How do you calculate the sum of terms once you know a_1 and d?

Find the last term using the general formula, take the average of the first and last terms, and multiply that average by the total number of terms: S_n = n * (a_1 + a_n) / 2.

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