explain the pattern below?

Question

Case 1: (15) (15)=225
Case 2: (25) (25)=625
Case 3: (35) (35)=1225
Case 4: (45) (45)=2025
Case 5: (55) (55)=3025
Case 6: (65) (65)=4225
Case 7: (75) (75)=5625
Case 8: (85) (85)=7225
Case 9: (95) (95)=9025

Answer 1

Rule: (Case Number) x (Case Number +1) & add 25 for last 2 digits

CASE 1: 1X2 = 2 and add 25 for last 2 digits = 225
CASE 2: 2x3 = 6 and add 25 for last 2 digits = 625
CASE 3: 3X4 = 12 and add 25 for last 2 digits = 1225
.
.
CASE 9: 9X10=90 and add 25 for last 2 digits = 9025
.
.
Case 20: (205) (205) = 42025
CASE 20: 20X21 = 420 & add 25 for last 2 digits = 42025

A really nice pattern. You can solve some difficult problems in your head even. For example (9995)(9995)

(999)(1000) = 999000
add 25 for the last 2 digits to get 99,900,025.

Answer 2

Fibonacci series.

Eg:

15x15 = 225
25x25 = 625
and so on..

Let's represent the list by

15x15 = (1) x 2 x 10^2+ 5^2
25x25 = (3) x 2 x 10^2+ 5^2
35x35 = (6) x 2 x 10^2+ 5^2
45x45 = (10)x 2 x 10^2+ 5^2
55x55 = (15)x 2 x 10^2+ 5^2

Noting the numbers in brackets
3-1 =2
6-3 =3
10-6=4
15-10 = 5

Long story short.. to find out what (105)(105) is,
add 5+4+3+2+1, multiply by 2 and then place "25" in the right most digits.

So (105)(105) = (10+9+8+7+6+5+4+3+2+1)x2 x 100 +25
= (110)100+25
= 11025

Answer 3

Multiplication by 10 more each time. Only the result is not 100 more.
Case 10:(105)(105)=11025

Answer 4

the numbers in the brackets (15) (15) are times together to get the answer 225 so if you times the number in each bracket by the one next to it, you get the number after the equals sign!!
check it on a calculator if you don't belive me!!!

Answer 5

subtract the answer from each other as you go up and see what you get. I pretty sure you already know that it's showing the number multiplying itself.

Answer 6

you are adding
400 to get 2 the second
600 to get too the third
800 to get to the forth
and sooo on

Answer 7

f[(m,5),(n,5)] = 10m(n + 1) + 25