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Consider the two arithmetic sequences:

$3,7,11,\ldots,603$

$2,9,16,\ldots,709$

How many terms are common to both?

Clearly the $n$th term of the first sequence is $a_n=4n-1$ and $b_n=7n-5$ for the second, but I am lost as to how to compute the total number of terms that overlap. Any ideas?

2 Answers 2

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In order for it to be the case that for certain integers $m,\,n$

$$ a_m=4m-1=7n-5=b_n\tag{1}$$

it must be true that

\begin{equation}7n-4m=4 \end{equation}

Thus it must be the case that $n=4k$ for some integer $k$.

So only elements $b_n=b_{4k} =28k-5$ of the $b_n$ sequence will occur in the $a_m$ sequence and it must be the case that $28k-5\le603$. Thus $1\le k\le21.$

So the $21$ numbers $23,51,79,\cdots,583$ will occur in both sequences.

But are these the only terms in common?

Putting $n=4k$ into equation $(1)$ we find that

\begin{eqnarray} 28k-4m&=&4\\ 7k&=&m+1 \end{eqnarray}

So $m+1$ must be a multiple of $7$. So we re-write the equation for $a_m$ as

\begin{eqnarray} a_m&=&4m-1\\ &=&4(m+1)-5\\ &=&28k-5 \end{eqnarray}

Thus the $21$ numbers of the form $28k-5$ are the only numbers the two sequences have in common.

In general we have $a_{7k-1}=b_{4k}$ for $1\le k\le21$.

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Simply check when $a_n $ can be equal to $b_n $. Beware, though, that they may overlap without it being in the same position. That is, do not check $4n - 1= 7n - 5$ ,check instead

$$a_n = b_k \iff 4n - 1 = 7k - 5$$

That should not be too difficult given that $n $ and $k $ must be integers.

We show it can be done:

$$4n - 1 = 7k - 5 \iff\\ 4n + 4 = 7k \iff\\ 4(n+1) = 7k \iff\\ 4j = 7k$$

Dividing left side by seven and right side by four, we get $\frac{4j}{7} = k$ and $j = \frac{7k}{4}$ from which we conclude that $j$ is a multiple of $7$ and $k$ is a multiple of $4$. Let us write $k = 4i$ and substitute:

$$4j = 7(4i) \iff\\ j = 7i$$

Thus if we pick some integer value for $i$, we then get $j = 7i, k = 4i, n = j-1$. All you have to do now is check the ranges for legal values of $i$.

Some common terms would be:

  • $i = 1, j = 7, k = 4, n = 6: a_n = 23; b_k = 23$

  • $i = 200, j = 1400, k = 800, n = 1399: a_n = 5595; b_k = 5595$

So clearly $i$ can't go all the way up to $200$. Can you tell what is its maximum value?

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    Right, I know they do not have to be in the same position. How does it help to simply check when the sequences are equal? That *is* the question. – 2017-01-12
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    @interrogative You might be able to write a script in MATLAB to figure the answer out. But I'm guessing there's a more elegant mathematical solution. – 2017-01-12
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    @user46944 I imagine it involves diophantine equations but I'm not really sure how to apply them here. – 2017-01-12
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    @interrogative check my answer now, please. I believe I made myself clearer. – 2017-01-12