等差数列

Two sequences are identical when every element in the sequence is the same:

But, checking every element would be tedious.

Is there a more efficient way? $\\$

If the same formula describes both sequences then they're identical.

We have two identical arithmetic sequences:

They're identical since they have the same initial term and common difference.

Recall that the formula for the $n^{\text{th}}$ term in an arithmetic sequence is $$a_n=a_1+d(n-1).$$ When two arithmetic sequences have identical $a_1$ (initial term) and $d$ (common difference), their $n^{th}$ term must be identical. Therefore, the two sequences are identical.

We have two identical geometric sequences:

Similar to arithmetic sequences, two geometric sequences are identical when they have the same initial term and common ratio.

Recall that the formula to the $n^{th}$ term in an geometric sequence is $$a_n=a_1r^{(n-1)}.$$ When two geometric sequences have identical $a_1$ (initial term) and $r$ (common ratio), their $n^{\text{th}}$ term must be identical. Therefore, the two sequences are identical.

Note that you won't always be given the initial terms and common differences/ratios of two sequences when determining whether they are identical. Try to figure out what they are from the given information, and work from there!

Try the problem below.

We have two arithmetic sequences. The $5$th, $6$th and $7$th terms of sequence A are $15$, $20$, $25$. The initial term of sequence B is $-5$ and has a common difference of $5$.

With the three terms from sequence A given, we can figure out its initial term and common difference—both are the same as sequence B's. Therefore, the two sequences are identical!

Identical Arithmetic Sequences: some examples $\\$

Two arithmetic sequences are identical when they:

• have the same initial term
• follow the same pattern

Example

What if two arithmetic sequences have the same pattern, but not the same initial term?

Both sequence A and sequence B are arithmetic sequences. They have the same common difference $5$.

Even thought A and B have the same common difference, they are NOT necessarily the same. Think about this:

What if sequence A starts with $2$ and sequence B starts with $5$?

A: $2, 7, 12, 17,22,\dots$

B: $5, 10, 15, 20, 25,\dots$

If we want two sequences to be identical, they must have both the same initial term (which this sequence doesn't have) and the same pattern (which this sequence has). Therefore, Sequence A and Sequence B are not identical sequences.

Let's try another example.

That's right! Identical sequences must have the same initial term and the same pattern in order to be identical.

Whenever you're looking at multiple sequences, even if you're looking at $5$ sequences, they must always have the same initial term and the same pattern in order for them all to be considered identical to each other.

Outside of math, you may encounter this concept when you're learning about DNA and RNA. DNA and RNA sequence alignment is important to determine if there is any overlap, which may give us clues about our evolutionary history!

Example Problem

Let's work through a common style of problem.

We have two sequences but only know some of the terms in each. $$\begin{align} S_1\quad&\begin{cases} a_{10}=44\\ a_{11}=48\\ a_{12}=52 \end{cases}\\ \\ S_2\quad&\begin{cases} a_3=16\\ a_8=36\\ a_{304}=1220 \end{cases} \end{align}$$ Are these sequences identical?

Well, it's probably not apparent from first glance. But, we can use math to figure it out!

For two sequences to be identical, they need to have the same formula to describe them. In other words, if they have the same first term and common difference or ratio, then they're the same sequence.

Step 1: Find the common difference or ratio for each sequence.

Looking at the first sequence, we can see each term increases by $4$. This means it's an arithmetic sequence with a common difference of $d=4$.

Looking at the second sequence, it isn't as obvious whether this is arithmetic or geometric. Instead of testing for both, let's assume it's arithmetic. If we get the same common difference between each terms then we know we were correct. If we don't, then we know it's not an arithmetic sequence and therefore, it cannot be identical to the first. Let's plug in some values! $$\begin{align} a_8&=a_3+(8-3)d\\ a_8&=a_3+5d\\ 36&=16+5d\\ 20&=5d\\ 4&=d\\ \\ a_{304}&=a_8+(304-8)d\\ a_{304}&=a_8+296d\\ 1220&=36+296d\\ 1184&=296d\\ 4&=d \end{align}$$ Since both these equations gave a common difference of $4$, we know the second sequence is arithmetic where $d=4$.

Hey! Both sequences have the same common difference! We're one step closer.

Step 2: Find the first term of each sequence.

Now that we know the common difference, we can use our arithmetic general formula to find the first term of each sequence. $$a_n=a_1+(n-1)d$$

Let's start with the first. $$\begin{align} a_n&=a_1+(n-1)d\\ a_{10}&=a_1+(10-1)(4)\\ a_{10}&=a_1+36\\ 44&=a_1+36\\ 8&=a_1 \end{align}$$

And now, the second. $$\begin{align} a_n&=a_1+(n-1)d\\ a_{3}&=a_1+(3-1)(4)\\ a_{3}&=a_1+8\\ 16&=a_1+8\\ 8&=a_1 \end{align}$$ They both have the same first term!

Since both sequences have the same first term and the same common difference, we know that they're identical!

(Video) Identical Sequences

by: Anusha Rahman

This video by Anusha Rahman goes over how to determine whether two sequences are identical.

Summary

Let's start by defining what identical sequences are:

Identical sequences: sequences in which all elements are identical

Identical arithmetic sequences: arithmetic sequences with the same initial term and common difference $$a_n = a_1 + d(n-1)$$

Identical geometric sequences: geometric sequences with the same initial term and common ratio $$a_n=a_1r^{(n-1)}$$

Example: Let's look at the following arithmetic sequences:

Sequence A

• 5th term = 50
• 7th term = 50

Sequence B

• 1st term = 10
• d=10

Are the two sequences identical?

Let's first look at the 5th term of Sequence A, creating an arithmetic sequence equation for it. $$a_n = a_1 + d(n-1)$$ $$50=a_1 + d(4)$$ $$50=a_1+4d$$

Now, let's look at the 7th term of Sequence A, creating an arithmetic sequence for it. $$a_n = a_1 + d(n-1)$$ $$70=a_1 + d(6)$$ $$70=a_1+6d$$

Using the substitution method to solve a linear system of equations, we will find that:

$$d=10$$ $$a_1=10$$

That means that our arithmetic equation for Sequence A is: $$a_n=10+10(n-1)$$

Now, let's look at Sequence B. We've already been given the initial term and the common difference. So, we can plug these into the formula for an arithmetic sequence: $$a_n=10 + 10(n-1)$$

The two formulas for Sequence A and Sequence B are the same! Therefore, we can say that the two sequences are identical.