Find 𝑛, given 𝑎𝑛 is equal to four 𝑛 plus five and 𝑎𝑛 is equal to 237.
What we’re actually looking at here is an expression for an arithmetic sequence. And we know that it’s an arithmetic sequence because an arithmetic sequence has a common difference. And this expression here and this sequence would have a common difference of four, as it’s the coefficient of 𝑛 which is the term number. I can demonstrate this quickly by showing you. This is the first term, so 𝑎 one, so our first term where we’d substitute in that 𝑛 is equal to one, would be four times one plus five which gives us nine. So the first term would be nine. And our second term would be equal to four multiplied by two, cause we’ve substituted 𝑛 is equal to two because it’s the second term, plus five which would be equal to 13. And therefore, our second term minus our first term is equal to 13 minus nine which is equal to four which is the same as the common difference we saw as the coefficient of 𝑛.
Okay, great. So now we kind of gained understanding of what this is. We can now find what 𝑛 is. Okay, so we’ve got 𝑎𝑛 is equal to four 𝑛 plus five. First of all, we’re gonna substitute our value for 𝑎𝑛 in, which gives us 237 is equal to four 𝑛 plus five. So now we solve for 𝑛. The first thing we do to do that is subtract five from both sides which gives us 232 is equal to four 𝑛. Now I’ve just rewritten this with the 𝑛s on the left-hand side.
And then finally, we’re gonna divide both sides by four which gives us 𝑛 is equal to 58, which means we know that the 58th term is equal to 237.