Question

Look at the pattern. What is the middle number in the 9th row. 1 1 1 1 2 1 1 3 3 3 1 1 4 6 4 6 4 1 1 1​

Answer 1

Step-by-step explanation:

the pattern is 31

hope it helps u

Answer 2

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

ANSWER:----

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, FN = (N-1)(N) + 1

For any Row N, the Last number, LN = (N)(N+1) - 1

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

[Note that FN – 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)*(FN - 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

I HOPE IT HELPFUL FOR YOU

THANKS