Explicit Formula (i think)

Question

Tom saves $1 the first day of the month, $2 the second day, $4 the third day, and so on. He continues to double his savings each day. Find the amount that he will save of the fifteenth day.

Answer 1

You’re right! There is a formula, and it’s very, very important !

Also, the problem has is slightly confusing (see note later).

Observe the 1 = 2^0     (or for some computer software, 2**0, or    2^1-1= 2⁰ )

   And that 2^1 =2

        2^2 = 4   (third day)

    . . .

      2^(n-1)    on the nth day

 

With computers, a similar formula tells you immediately how large an integer memory will contain or how many address of memory may be referenced by an instruction.  (It’s why memory values are a power of 2).  For example, a 64-bit computer may address (2^64) -1   (not quite the same formula) locations.

 

Now, the note:  The question asks for the amount that Tom saves on the fifteenth day.  Does that mean the amount that he saves just that day, or does it mean the total amount of his savings so far that month? (if he hasn’t spent any yet)

Answer 2

X = 2^(Day -1)

If Day = 1, the X = 2^0 = 1

if Day = 2, then X = 2^1 = 2

if Day = 3, then  X = 2^2 = 4

...

if Day = 15, then X = 2^14 = 16384

So on Day 15 Tom saves $16,234

Answer 3

Savings, S= 1+2+4=8+16+...2^14

 

S=1 + 2(1+2+4+8+16 ...+2^13)

 

S=1+2(S-2^14)

 

S=2^15 -1

 

 

Note 1=2^1 -1

    1+2=2^2 -1

1+2+4=2^3 -1  etc.