# Question Video: Finding the Arithmetic Sequence given the Value of One of Its Terms Mathematics

Find the arithmetic sequence given 𝑎₁ = 13 and 𝑎_(18𝑛) = 18𝑎_(𝑛).

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### Video Transcript

Find the arithmetic sequence given 𝑎 sub one equals 13 and 𝑎 sub 18𝑛 is equal to 18 multiplied by 𝑎 sub 𝑛.

We begin by recalling that an arithmetic sequence is a sequence with a common difference between consecutive terms. The general term or 𝑛th term of an arithmetic sequence can be found using the formula 𝑎 sub 𝑛 is equal to 𝑎 sub one plus 𝑛 minus one multiplied by 𝑑, where 𝑎 sub one is the first term of the sequence and 𝑑 is the common difference. In this question, we are told that the first term 𝑎 sub one is equal to 13. And we are also told that 𝑎 sub 18𝑛 is equal to 18 multiplied by 𝑎 sub 𝑛.

Using the general formula, 𝑎 sub 18𝑛 is equal to 𝑎 sub one plus 18𝑛 minus one multiplied by 𝑑, and 18 multiplied by 𝑎 sub 𝑛 is equal to 18 multiplied by 𝑎 sub one plus 𝑛 minus one multiplied by 𝑑. We can replace 𝑎 sub one with 13 and set the two expressions on the right-hand side of the equations equal to one another. We have 13 plus 18𝑛 minus one multiplied by 𝑑 is equal to 18 multiplied by 13 plus 𝑛 minus one multiplied by 𝑑. Distributing the parentheses or expanding the brackets on the left-hand side gives us 18𝑛𝑑 minus 𝑑. On the right-hand side, 𝑛 minus one multiplied by 𝑑 is equal to 𝑛𝑑 minus 𝑑.

We can then multiply the expression in the square brackets by 18. This gives us 234 plus 18𝑛𝑑 minus 18𝑑. This is equal to 13 plus 18𝑛𝑑 minus 𝑑. We can subtract 18𝑛𝑑 from both sides of the equation. This gives us 13 minus 𝑑 is equal to 234 minus 18𝑑. We can then add 18𝑑 and subtract 13 from both sides of the equation. 17𝑑 is therefore equal to 221. Dividing both sides of this equation by 17 gives us 𝑑 equals 13. The common difference of the arithmetic sequence is 13.

Since the first term of the sequence was 13, we can add the common difference to this to calculate the second term. This is equal to 26. The third term of the sequence is 39. We can therefore conclude that the arithmetic sequence with first term 13, where 𝑎 sub 18𝑛 is equal to 18 multiplied by 𝑎 sub 𝑛 is 13, 26, 39, and so on.