Question

Study the pattern:

$$\begin{gathered} 1\times 8+1=9 \\ 12\times 8+2=98 \\ 123\times 8+3=987 \\ 1234\times 8+4=9876 \\ 12345\times 8+5=98765 \end{gathered}$$

Write the next two steps. Can you say how the pattern works? (Hint: $12345=11111+1111+111+11+1$)

Answer 1

Step 1: Understand the given pattern:

The given pattern is shown below:

$$\begin{gathered} 1\times 8+1=9 \\ 12\times 8+2=98 \\ 123\times 8+3=987 \\ 1234\times 8+4=9876 \\ 12345\times 8+5=98765 \end{gathered}$$

We observe on the left-hand side the pattern of the numbers is $(1,12,123,1234,12345)$ which means the digits are consecutively increasing . These numbers are being multiplied by $8$ in every step. After this, they are added to consecutive numbers $(1,2,3,4,5)$. While on the right-hand side the pattern of the numbers is $(9,98,987,9876,98765)$ which means the digits are consecutively decreasing.

Step 2: Finding the next two steps:

With the help of the given pattern, we can write the next step as,

$123456\times 8+6=987654$

This can be evaluated as,

$$\begin{gathered} 123456=\left(111111+11111+1111+111+11+1\right) \\ 123456\times 8=\left(111111+11111+1111+111+11+1\right)\times 8 \\ 123456\times 8=888888+88888+8888+888+88+8 \\ 123456\times 8=987648 \\ 123456\times 8+6=987648+6 \\ 123456\times 8+6=987654 \end{gathered}$$

Similarly, $1234567\times 8+7=9876543$

$$\begin{gathered} 1234567=\left(1111111+111111+11111+1111+111+11+1\right) \\ 1234567\times 8=\left(1111111+111111+11111+1111+111+11+1\right)\times 8 \\ 1234567\times 8=8888888+888888+88888+8888+888+88+8 \\ 1234567\times 8=9876536 \\ 1234567\times 8+7=9876536+7 \\ 1234567\times 8+7=9876543 \end{gathered}$$

Therefore the next two steps are $123456\times 8+6=987654$ and $1234567\times 8+7=9876543$.