Enormous Sum

Compute the sum of the sum of all 8-digit numbers and the sum of all 10-digit numbers.
Source: mathcontest.olemiss.edu 11/3/2008

SOLUTION

We need to recall the formula used to calculate the sum of an arbitrary sequence of counting numbers. Given a sequence of counting numbers a_1,a_2,a_3,\cdots,a_n arranged in increasing order starting from the smallest number a_1 to the largest number a_n:

a_1+a_2+a_3+\cdots+a_n=\frac{\left (a_n+a_1\right )\left (a_n-a_1+1\right )}{2}
 
You may have seen this formula when the sequence is 1,2,3,\cdots,n:

1+2+3+\cdots+n=\frac{\left (n+1\right )n}{2}

STEP 1 Calculate the sum S_8 of all 8-digit numbers.

We will calculate by hand because we want exact values and not rounded up approximation given by the calculator. Remember we do not count leading zeros as part of the numbers, i.e. 00000001 is not considered an 8-digit number.

The smallest 8-digit number is a_1=10,000,000=10^7.

The largest 8-digit number is a_n=99,999,999.

Applying the above formula:

S_8=\frac{\left (99,999,999+10,000,000\right )\left (99,999,999-10,000,000+1\right )}{2}

=\frac{\left (109,999,999\right )\left (10^8-10^7\right )}{2}

Factoring out 10^7:

=\frac{\left (109,999,999\right )\left (10^7\right )\left (10-1\right )}{2}

=\frac{\left (109,999,999\right )\left (10^7\right )\left (9\right )}{2}

=109,999,999\times45\times10^6

Let’s calculate the product 109,999,999\times45 using pencil and paper as follows:

109,999,999\times45=109,999,999\left (5+40\right )

=549,999,995+4,399,999,960

=4,949,999,955

Thus, S_8=4,949,999,955\times10^6.

STEP 2 Calculate the sum S_{10} of all 10-digit numbers.
The smallest 10-digit number is a_1=10^9.

The largest 10-digit number is a_n=9,999,999,999.

S_{10}=\frac{\left (9,999,999,999+10^9\right )\left (9,999,999,999-10^9+1\right )}{2}

=\frac{\left (10,999,999,999\right )\left (10^{10}-10^9\right )}{2}

Factoring out 10^9:

=\frac{\left (10,999,999,999\right )\left (10^9\right )\left (10-1\right )}{2}

=\frac{\left (10,999,999,999\right )\left (10^9\right )\left (9\right )}{2}

=10,999,999,999\times45\times10^8

Let’s calculate the product 10,999,999,999\times45 using pencil and paper as follows:

10,999,999,999\times45=10,999,999,999\left (5+40\right )

=54,999,999,995+439,999,999,960

=494,999,999,955

Thus, S_{10}=494,999,999,955\times10^8.

Finally, we calculate the enormous sum of S_8+S_{10}:

          4,949,999,955,000,000
49,499,999,995,500,000,000
49,504,949,995,455,000,000

Answer: 49,504,949,995,455,000,000.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , . Bookmark the permalink.

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