Question

Evaluate the following pattern and provide step by step solution about how the pattern works. Also find out the next four steps of this pattern.
$1 \times 8 + 1 = 9$
$12 \times 8 + 2 = 98$
$123 \times 8 + 3 = 987$
$1234 \times 8 + 4 = 9876$
$12345 \times + 5 = 98765$

Answer 1

Hint: First we will study the pattern carefully .By checking the left side we will find an increment in the digits. Using logic we will find out that the first term is 1 and in the second term 11+1=12 and this pattern continues such that in the last term it Is $11111 + 1111 + 111 + 11 + 1 = 12345$.

Complete step by step solution:
We can write $1 \times 8 + 1$as
$1 \times 8 + 1 = $\[\left( {1 \times {{10}^{1 - 1}}} \right) \times 8 + 1 = 9\]
Similarly we can write
$12 \times 8 + 2 = (11 + 1) \times 8 + 2$
\[ = \left( {1 \times {{10}^{2 - 1}} + 2 \times {{10}^{2 - 2}}} \right) \times 8 + 2 = 98\]
In this step we can write
$123 \times 8 + 3 = (111 + 11 + 1) \times 8 + 3$
=\[ = \left( {1 \times {{10}^{3 - 1}} + 2 \times {{10}^{3 - 2}} + 3 \times {{10}^{3 - 3}}} \right) \times 8 + 3 = 987\]
Similarly we write
$1234 \times 8 + 4 = (1111 + 111 + 11 + 1) \times 8 + 4$
 \[ = \left( {1 \times {{10}^{4 - 1}} + 2 \times {{10}^{4 - 2}} + 3 \times {{10}^{4 - 3}} + 4 \times {{10}^{4 - 4}}} \right) \times 8 + 4 = 9876\]

We write
$12345 \times 8 + 5 = (11111 + 1111 + 111 + 11 + 1) \times 8 + 5$
 \[ = \left( {1 \times {{10}^{5 - 1}} + 2 \times {{10}^{5 - 2}} + 3 \times {{10}^{5 - 3}} + 4 \times {{10}^{5 - 4}} + 5 \times {{10}^{5 - 5}}} \right) \times 8 + 5 = 98765\]
So, observing the similarity of the pattern we can say that
\[\left( {1 \times {{10}^{n - 1}} + 2 \times {{10}^{n - 2}} + 3 \times {{10}^{n - 3}} + ................ + n \times {{10}^{n - n}}} \right) \times 8 + 5 = \left( {\sum\limits_{i = 1}^n {i \times {{10}^{n - i}}} } \right) \times 8 + n\]
Thus, following the similarity of the pattern we can write the next four steps
$123456 \times 8 + 6 = (111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 6$$$ $$ \[ = \left( {1 \times {{10}^{6 - 1}} + 2 \times {{10}^{6 - 2}} + 3 \times {{10}^{6 - 3}} + ................ + 6 \times {{10}^{6 - 6}}} \right) \times 8 + 6 = 123456 \times 8 + 6 = 987654\]
Similarly in next step
$1234567 \times 8 + 7 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 7 = 9876543$
In the next step
$12345678 \times 8 + 8 = (11111111 + 1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 8 = 98765432$
 Similarly we can write in the next step
 $123456789 \times 8 + 9 = (111111111 + 11111111 + 1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 9 = 987654321$
Thus we can find the next four steps.

Note: In Mathematics, number patterns are the patterns in which the numbers follow a certain similarity or common relationship. Example: $1,5,10,15, \ldots $in this pattern every term is multiple of $5$. The first task is to find the common similarity among the terms.