Question Video: Using Arithmetic Sequences in a Real-World Context Mathematics

Mrs. Miller has $3,250 in her bank account. If she plans to deposit $100 per month, write the first 6 terms of a sequence that would represent her monthly account balances starting with her current balance as the first term.

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Video Transcript

Mrs. Miller has 3,250 dollars in her bank account. If she plans to deposit 100 dollars per month, write the first six terms of a sequence that would represent her monthly account balances, starting with her current balance as the first term.

So, we’ve been asked to find the first six terms of the sequence that represents Mrs. Miller’s monthly account balances. We’re told that she has 3,250 dollars in her bank account initially. So this, her current balance, is the first of our six terms. We’re then told that Mrs. Miller plans to deposit 100 dollars per month into her account. This means that her balance will increase by 100 dollars every month. So the second term of our sequence will be 100 more than the first term. It’s 3,250 plus 100, which is 3,350. To find the third term, we add 100 to the second term, 3,350 plus 100, which is 3,450. And we can then continue in this way to find the remaining three terms. They are 3,550, 3,650, and 3,750. In fact, these terms form an arithmetic sequence because the differences between each pair of successive terms are constant. As we move from one term to the next, we always add the same amount, which is 100 dollars.

Although we aren’t asked to, we could write down a formula for the general term in this sequence. We recall that the general term in an arithmetic sequence can be expressed as 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑, where 𝑎 sub 𝑛 means the 𝑛th term in the sequence. 𝑎, or sometimes 𝑎 sub one, denotes the first term in the sequence. 𝑛 denotes the term number. And 𝑑 represents the common difference between the terms. The first term for this sequence is 3,250, and the common difference 𝑑 is 100.

We could now use this formula to calculate any term in our sequence. So, for example, to calculate the third term, we substitute 𝑛 equals three, which gives 𝑎 sub three is equal to 3,250 plus 100 multiplied by three minus one. That’s 3,250 plus 200, which is 3,450. And this agrees with the value we’ve already found for the third term in the sequence. We could use this general rule to calculate any term in the sequence if we wish.

So, we’ve found that the first six terms of the sequence representing Mrs. Miller’s monthly account balances are 3,250, 3,350, 3,450, 3,550, 3,650, and 3,750, where all of these represent an amount in dollars.

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