Sequences and series help please?

Question

Gerry owns a milking cow. On the first day he milks the cow, it produces 600 ml of milk. On each successive day, the amount of milk increases by 10%.

How much milk does the cow produce on the seventh day?

Answer 1

Fantasy is passing if you don't do your homework.

My secret is I did my homework.

10% added is 100% + 10% = 1.1

day one x

day 2 x*1.1

day 3 x*1.1 * 1.1

day 4 x*1.1* 1.1 * 1.1

or day w ==> x * (1.1)^(w-1)  

it's (w-1) as only 100% the first day

so w = 7 milk = 600 * (1.1)(1.1)..(1.1) until 6 brackets = 600 * (1.1)^6

Answer 2

960ml

Answer 3

Your answer done by simply adding the days:

Day 1: 600ml

Day 2: 600*.10 +600ml= 660ml  

Day 3: 660*.10 +660ml= 726ml  

clearly, a pattern is developing, which is:

milk production = n

n = previous day*.1 + previous day

Day 4: 726*.10 + 726ml = 798.6ml

Day 5: 798.6*.10 + 798.6ml = 878.46ml

Now the question comes, is there any way to come up with a formula that figures out Day 7, without needing the previous day, Day 6. The answer is yes.

This problem is like an interest rate problem, i.e. if you change the milk to dollars, it is like Gerry is earning 10% interest on his $600 dollar investment each day. Put into math, we derive this formula:

FV =PV (1+i)^n where Your future value (FV) is found by taking the current present value (PV, i.e. 600) and multiplying it by the interest rate taken to the nth day.

Put into easier to understand lingo, the formula that makes it so that you don't have to add every day up is:

VARIABLES:

MP, where MP = Milk Produced

N, where N = Day in question, (i.e. the Seventh Day for the purpose of     this problem.)

I, where I = Daily Increase Percentage.

Those variables create this formula:

MP = 600ml (1+I)^(N-1)

And now we can find your answer much faster!!

Day 6: MP = 600 * (1 + .1)^(6-1) = 600* 1.1^5 = 600* 1.61051 = 966.306ml of milk

Day 7: MP = 600* (1.1)^6 = 600*1.771561 =  1026.9366ml

Thus, your final answer (rounded up) is 1027ml's of milk is produced on the seventh day.

Hopefully, you now see two ways to solve the problem, and clearly, the formula is much faster than the other. Good luck with your future math.

Answer 4

1 = 600

2 = 660

3 = 726

4 = 798.6

5 = 878.46

6 = 966.306

7 = 1062.9366 ml

Answer 5

So day 1 => 600 ml

Then Day 2 => 600 * 1.1 = 600*(1.1)^1

Day 3 => 600*1.1*1.1 = 600*(1.1)^2

And so on.

So Day 7 => 600*(1.1)^6 = 1062.9366 ml or 1.06 liters