Given the following sequence of numbers:
2, 4, 6, 8, 10, …

What mathematical formula best describes the sequence above?
One answer is 2n, where n is a positive integer beginning with n = 1. So,
If n = 1, then 2(1) = 2, the first term in the sequence.
If n = 2, then 2(2) = 4, the second term in the sequence.
If n = 3, then 2 (3) = 6, then third term in the sequence.
And so on and so forth.

One way to obtain this formula 2n is through a finite difference method. First, obtain a common difference d. To compute for d, subtract a preceeding term from a succeeding term.

4 – 2 = 2, 6 – 4 = 2, 8 – 6 = 4. Here d = 2.

Since there is a common difference d, then the correct formula is expressed in the form 2n, where n is an integer that starts with 1.

Sometimes you need to make adjustments with the formula. Consider, for instance, the sequence below:

2, 5, 8, 11, 14, …

Here the common difference d is 3 since 5 – 2 = 3, 8 – 5 = 3, 11 – 8 = 3, etc.

So part of the formula should look like 3n, where n is an integer that begins with 1. Since the first term in the sequence is 2, you will need to make an adjustment.

If n = 1, then 3(1) = 3. But 3 - 1 = 2, which gives the correct first term in the sequence.

Next, try the formula 3n – 1 for the second term:

If n = 2, then 3(2) – 1 = 6 – 1 = 5, the second term in the sequence.

Try the formula again to see if it works for the third term:

If n = 3, then 3(3) – 1 = 9 – 1 = 8, the third term in the sequence.

Hence, the correct formula for the sequence 2, 5, 8, 11, 14, … is 3n – 1.

Visual Algebraic Approach:

The method of finite differences is a useful generalization strategy. For most students, using this strategy should not be very difficult. With students with learning disabilities in math, explaining the process can be quite overwhelming for them. One visual generalization strategy is to start out with sequences of numbers that are expressed in figures. For instance, consider Problem 1 below which asks students to determine a formula for the total number of toothpicks that are needed to form n adjacent squares. Teachers of students with learning disabilities in math can employ a visual generalization strategy in which students with learning disabilities in math can see what is actually taking place when they are formulating a generalization.

Figure Number 1 2 3
Number of Toothpicks
used to form the squares 4 7 10

In Problem 1 above, note the sequence of numbers that matters is 4, 7, 10. Of course, with students with learning disabilities in math, a sequence of five or more figures might be necessary for them to see a pattern. The following steps can be used with students with learning disabilities in math:

1. Ask students with learning disabilities in math to first count the number of toothpicks formed by each figure.

2. After they have established the numbers 4, 7, and 10, teach them what a common difference is. For this example, verify with them that the common difference d is 3 since 7 – 4 = 3 and 10 –7 = 3.

3. Then, just like in the explanation provided above, tell the class that a partial formula might take the form 3n, where n is an integer beginning with 1.

4. Link the formula 3n with the sequence 4, 7, 10. Since the first term is 4, then there is a need to add 1 to the form 3n. So 3n + 1 is a correct formula.

5. Ask students with learning disabilities in math to verify at least for the first three cases:

If n = 1: 3(1) + 1 = 3 + 1 = 4, the first term.
If n = 2: 3(2) + 1 = 6 + 1 = 7, the second term.
If n = 3: 3(3) + 1 = 9 + 1 = 10, the third term.

A good visual generalization strategy is to ask students with learning disabilities in math what is happening with the squares at each stage in terms of the needed number of toothpicks to form them. For instance, teachers of students with learning disabilities in math could point out that 4 sticks are needed to form the first square, an additional 3 sticks are needed to form the second figure, which makes 7, and 3 more sticks when added to the second figure will form the third figure, producing 10. Hence, another formula is 4 + 3n. But in this case, n starts with 0 since no sticks are added at the start of the sequence.

When students with learning disabilities in math are able to establish a visual generalization by seeing relationships visually, then teachers need to provide them with similar exercises until they feel comfortable in using the strategy. Teachers should not forget to emphasize what the variable “n” stands for as it might start from 1 or 0 depending on how the pattern is analyzed.

Use of Frames for Visual Generalization:

In the case of geometry, teachers of students with learning disabilities in math could present a sequence of pictures – called frames - of geometric objects or other figures in which a known, common property is clearly visible. students with learning disabilities in math are then led to describe the property and state it in their own words.

For instance, in geometry, students can be presented with the medians of five to ten different triangles. (A median is a segment drawn from one vertex of a triangle to the midpoint of the opposite side of the triangle.) The teacher then points out the medians (calling them “segments” first) in each case and asks the students to make an observation or observations about what is or are common to all the segments.

As a follow-up activity, students can be given a worksheet activity in which they will decide whether a geometric object has the property just discussed.

The teacher can also use flash cards with pictures of geometric objects that have or do not have the property. Students should be encouraged to respond or explain orally.