 |
Siseko is hoping to buy a car when he turns 18. His grandmother offers to sell him her car for R131 071. He works on Saturdays only and earns R200 every Saturday. He save's R1 on the first Saturday. Each Saturday thereafter he doubles the amount he saved on the previous Saturday. He puts the money into a bottle on his desk and not in a bank account.
In which week will his weekly payment exceed his earnings for the first time and what is the amount of the final payment he can afford?
He saves R1 on the first Saturday S in, say, week 0.
w.0 = 2^0 = 1
Each Saturday thereafter he doubles (hence the base 2) the amount he saved on the previous Saturday. so in week 1
w.1 = 2^1
And he goes on:
w.2 = 2^2
...
w.7 = 2^7 = 128
Well, we can sum up what he has saved until then:
Generally this sum can be expressed as:
sum 2^k, for k=0 to n equals 2^(n+1) - 1
If n=7 this amounts to 2^8 - 1 = 255
On the 8th Saturday, in week 7, the final payment he can afford will be R128.
In the following week he would have to pay R256, which he can clearly not afford earning R200 per week.
On the 17th Saturday in week 16 he will save:
w16 = 2^16 = R65536.
His savings would then sum up to:
sum 2^k, for k=0 to 16 equals
2^17 -1 = R131071
Thank you
Just one more thing
If he had been able to afford the payment, how long would it have taken him to save R131071?