a. Find the middle number in the 99th row.

Question

Row 1 1
Row 2 3 5
Row 3 7 9 11
Row 4 13 15 17 19
Row 5 21 23 25 27 29
a. Find the middle number in the 99th row.
b. What is the sum of the numbers in the 30th row.
c. Find the difference between the first number in row 90 and the first number in row 89.

Answer 1

First, consider the fact that these are the odd integers.

Now, remember Gauss’ Formula for the Sum Of The Numbers From 1 To N: (N)(N+1)/2.

Note: this is how to make odd numbers:
                     write out the list like 
                 21 + 23 + 25 + 27 +29         [find sum, for example]

      5*20 + (1 +  3 +   5 +   7 +  9)     [subtract 20 from each number]

 

      5*20 + 1 + 2 + 3 + 4 + 5                 100+5*6/2

                    + 1 + 2 + 3 + 4                         4*5/2

                                                            ---------------

                                                                    125

Then, consider the questions. We will need to calculate the first number in Row N; this will allow us to calculate all of the answers.

There are (N-1)N/2 numbers before Row N. So, [using previous logic]
                     For any Row N, the First number, F_N = (N-1)(N) + 1
                     For any Row N, the Last number, L_N = (N)(N+1) - 1

c. Use the F_N formula to find the difference F₉₀ - F₈₉

              [Note that F_N – F(_N-1) = 2(N-1) (from above formulas]
                  F₉₀ – F₈₉ = 2(90-1) = 178

a. the middle number in any Row N (like the 99th) is the mean (arithmetic average) of the first and the last numbers in that row:

                (FN+LN)/2

             ( (98*99/2 + 1) + (99*100)/2 – 1) = 9801

b. To find the sum of the numbers in any row:

                So, [see above logic]

                 for Row N, this is: (N)*(F_N - 1) + (N)(N+1)/2 + (N-1)(N)/2

The Sum of the numbers in Row 30 is:
             (30)((29*30+1)-1) + (30)(31)/2 + (29)(30)/2

            = 30*29*30 + 30*31/2 + 29*30/2

            = 26100 + 465 + 435

            = 27000

Answer 2

Arrange the numbers in a pyramid

 

                      1
                   3   5
                7    9   11
            13  15  17  19
         21  23  25  27  29

 

Note the pattern by which each diagonal increases.

Hint: It is a sequence within a sequence

Use that information to answer the questions.

Answer 3

Here's part a) using different reasoning.

Row 5 starts with the 11th odd number. The previous rows have 4, 3, 2, and 1 odd numbers for a total of 10 odd numbers.

The 99th row will begin with which odd number ?

Add the number of odd numbers from each row.

1+2+3+4+...+95+96+97+98.

Add the first and last to get 99; add the second and next to last to get 99; and so on. You have 49 groups of 99.

99*49=4851 odd numbers that precede the first odd number in row 99.

The first odd number in row 99 is what number ?

Look for a pattern. The first odd number in row 5 is double the number of all the previous numbers plus 1.

2*4851+1=9703

9703 is the first odd number in row 99

the last number in row 99 is two times the number of the previous row added to the first number of the 99th row

2*98=196

9703+196=9899

the numbers go from 9703 to 9899

there is an odd number of numbers because 99 is odd

the middle number is the sum of the first and the last divided by 2 because 99 is odd and if you have an odd number of numbers then there is exactly one number in the middle; also these numbers are consecutive odd numbers so the first plus the last divided by 2 gives you the middle number

(9703+9899)/2=19,602/2=9801