Video Transcript
Mrs. Jones has 3250 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’re looking for the first six
terms of the sequence that represent the amount of money Mrs. Jones has in her
account each month. That’s her monthly account
balances. We’re told that we should use her
current balance as the first term in this sequence. And from the information given in
the question, we know that her current balance is 3250 dollars. So we can go ahead and write this
as the first term in our sequence.
We’re then told that Mrs. Jones
plans to deposit 100 dollars per month. So every month, her account balance
will be increasing by 100 dollars. This means that, to find the next
term in our sequence, each time we just need to add 100 to the previous term. We’re adding 100 to the balance the
month before. 3250 plus 100 is 3350. Adding 100 again gives 3450. And we can continue in this way to
find the remaining three terms: 3550, 3650, and 3750.
Now this is a reason to be a
straightforward problem. But there is actually another
method that we could use to calculate the terms in this sequence. As the amount that we’re adding
each time is constant, our sequence has what’s called a common difference. And it’s, therefore, an arithmetic
sequence.
There is a general formula that we
can use to find any term in an arithmetic sequence, if we know the first term and
the common difference. It’s this. 𝑇𝑛 — that’s the 𝑛th term — is
equal to 𝑎, the first term, plus 𝑛 minus one multiplied by 𝑑, the common
difference. In our sequence, the first term is
Mrs. Jones’s starting account balance. That’s 3250. And the common difference is
100. So we can use this formula to find
any term in the sequence.
So, for example, to find the sixth
term in the sequence, we substitute 𝑛 equals six, giving that the sixth term is
equal to 3250 plus six minus one multiplied by 100. That’s 3250 plus 500, which is
3750. And if we look at the sequence
we’ve already written down, we can see that this is indeed the sixth term. There wasn’t any particular need to
use this more formal method in this problem. But it would be useful if we’d been
asked to find, for example, the hundredth term, as we could do this without
calculating all of the previous 99.
So either by adding 100 each time
or by using the general formula for calculating the 𝑛th term of an arithmetic
sequence, we’ve found that the first six terms in the sequence representing Mrs.
Jones’ monthly account balances are 3250, 3350, 3450, 3550, 3650, and 3750.
