Yesterday, two friends went into a bank to open savings accounts. allie started by putting $783 in her account, and she will deposit an additional $27 each week. nick made no initial deposit, but he will add $288 more each week. in a few weeks, the friends will have the same account balance. what is that account balance? how many weeks will that take?

Both accounts of Nick and Allie will have same amount of $864 in time period of 4 weeks.

Solution:

Given, Yesterday two friends went into a bank to open savings accounts.  

Allie started by putting $783 in her account, and she will deposit an additional $27 each week.  

Nick made no initial deposit, but he will add $288 more each week.  

In a few weeks, the friends will have the same account balance.  

Here, account balance of Allie for every week forms an A.P as

783, 810, 837, ……. With common difference 27

And, account balance of nick for every week will be  

0, 288, 576, …….. with common difference 288

Let, after n weeks the balance in both accounts be equal.

Then, nth term of both series will be equal, we know that, nth term tn of A.P is a + (n – 1)d  

Where, a is first term, d is common difference and n is number of terms.

Now, nth term of allie series = nth term of nick series.

783 + (n – 1) x 27 = 0 + (n – 1) x 288  

Here, we used same n for both sides because n represents number of weeks and it is common to both

783 + 27n – 27 = 288n – 288

756 + 288 = 288n – 27n

261n = 1044

n = 4.

So, it will take 4 weeks to have same balance and now, substitute n value in nth term of nick series.

Hence, both accounts will have same amount of $864 in time period of 4 weeks

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.