## Step (1)

The table below shows

all the triangular numbers multiplied by 2

in the left column

&

the respective triangular numbers

generated from these

in the right column.

##### Triangular Numbers Table

##### Base

number

##### Triangular

number

0

(0 x 1)/2 = 0

2

(2 x 3)/2 = 3

6

(6 x 7)/2 = 21

12

(12 x 13)/2 = 78

20

(20 x 21)/2 = 210

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10(0) + 1 = 1

10(3) + 1 = 31

10(21) + 1 = 211

10(78) + 1 = 781

10(210) + 1 = 2101

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1 = 15

1 + 31 = 32 = 25

1 + 31 + 211 = 243 = 3 5

1 + 31 + 211 + 781 = 1024 = 45

1 + 31 + 211 + 781 + 2101 = 3125= 55

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one can deduce that in general

n to the 5th. power is

the summation of triangular numbers

per the formula shown below: