Question

Study the pattern:
$1\times 8+1=9$
$12\times 8+2=98$
$123\times 8+3=987$
$1234\times 8+4=9876$
$12345\times 8+5=98765$
Write the next four steps. Can you find out how the pattern works?

Answer 1

Hint: We notice that the left-sided multiplicand keeps on adding a new digit at the unit's place and the number on the right-hand side of the equation keeps adding a new digit to its unit place as well.

Complete step-by-step answer:
We notice that in each of the equations except for the first one, the left-sided multiplicand on the left-hand side of the equation gets left shifted by one place and adds a new digit to its unit place which is one more than the digit at the previous unit place.
This means that 123 in the previous equation becomes 1234 in the next equation.

The right-sided addend on the left-hand side of an equation becomes one more than the right-sided addend in the previous equation.
This means that 2 in the previous equation becomes 3 in the next equation.
Similarly, the number on the right-hand side of the equation gets left shifted by one place and adds a new digit to its unit place which is one less than the digit at the previous unit place.
This means that 987 in the previous equation becomes 9876 in the next equation.

Noticing this pattern in the existing equations, we produce the next four equations in the series which are as follows:
In the fifth step of the pattern we got 1,2,3,4,5 already, so in the sixth step of the equation it will be 1,2,3,4,5,6 and the consecutive addend will increase by 1.
Also the result in the fifth step is already 98765, in sixth step it will be 987654
$123456\times 8+6=987654$

Similarly we can get step 7,8 and step 9.
$1234567\times 8+7=9876543$
$12345678\times 8+8=98765432$
$123456789\times 8+9=987654321$

Note: The pattern should be carefully observed. It’s important not to miss the addend on the right side. If we miss out on increasing addend by 1 in the next step, we will get $123456\times 8+5\ne 987654$.