Question

\(l, m, n, o, p\) An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence? I. \(3 l, 3 m, 3 n, 3 o, 3 p\) II. \(\quad l^{2}, m^{2}, n^{2}, o^{2}, p^{2}\) III. \(l-5, m-5, n-5, o-5, p-5\)

Short answer

I and III are arithmetic sequences.

Step-by-step solution

Understanding an Arithmetic Sequence

An arithmetic sequence is defined as a sequence in which each term after the first is the sum of the previous term added to a constant difference. This means that the difference between consecutive terms is the same throughout the sequence.

Analyze Sequence I: 3l, 3m, 3n, 3o, 3p

Multiply each term of the arithmetic sequence by 3. This does not change the common difference. If the original sequence has a common difference of d, then the new sequence will have a common difference of 3d. Hence, this is still an arithmetic sequence.

Analyze Sequence II: l^2, m^2, n^2, o^2, p^2

Square each term of the arithmetic sequence. Squaring each term changes the common difference because the differences between squares of terms are not constant. Hence, this will not be an arithmetic sequence.

Analyze Sequence III: l-5, m-5, n-5, o-5, p-5

Subtract 5 from each term of the arithmetic sequence. Subtracting the same constant from each term does not change the common difference. Hence, the new sequence has the same common difference as the original one, making it an arithmetic sequence.

Key concepts

Common Difference

The foundation of any arithmetic sequence is the 'common difference'. This is the constant amount that you add to each term to get the next term. In other words, if you have a sequence where each term increases by the same value, that value is your common difference.
If our sequence is: l, m, n, o, p and it is an arithmetic sequence, then:
m - l = d
n - m = d
o - n = d
p - o = d
Here, d is the common difference. All these equalities show that subtracting one term from the next one gives the same value, which is d. This characteristic defines the sequence as arithmetic. Understanding this concept makes it easier to analyze the impact of different operations on the sequence, such as multiplication or adding/subtracting constants.

Multiplication of Sequences

When you multiply each term in an arithmetic sequence by a constant, the common difference is also multiplied by that constant. Let's consider our arithmetic sequence again: l, m, n, o, p with common difference d.
When we multiply each term by a constant, say 3, our new sequence becomes: 3l, 3m, 3n, 3o, 3p.
In this new sequence:

• (3m - 3l) = 3d
• (3n - 3m) = 3d
• (3o - 3n) = 3d
• (3p - 3o) = 3d

This shows that the new sequence retains its arithmetic properties, but the common difference has changed from d to 3d. Therefore, multiplying every term in an arithmetic sequence by a constant still results in an arithmetic sequence.

Effect of Constants on Sequences

Operations involving the addition or subtraction of a constant also have interesting effects on arithmetic sequences. Let's look at our base sequence: l, m, n, o, p with common difference d again.
If we subtract a constant, say 5, from every term, our new sequence becomes:
(l - 5), (m - 5), (n - 5), (o - 5), (p - 5).
In this sequence:

• ((m - 5) - (l - 5)) = m - l = d
• ((n - 5) - (m - 5)) = n - m = d
• ((o - 5) - (n - 5)) = o - n = d
• ((p - 5) - (o - 5)) = p - o = d

This indicates that subtracting the same constant from each term does not change the common difference. The sequence remains arithmetic because the difference between any two consecutive terms is still d.
This principle applies similarly if you add a constant. Adding or subtracting the same value from each term shifts the entire sequence up or down without affecting the common difference.