Explain the Pattern - Part 1

Algebra Level 1

1 × 8 = 8 11 × 88 = 968 111 × 888 = 98568 1111 × 8888 = 9874568 11111 × 88888 = 987634568 111111 × 888888 = 98765234568 \begin{array} { r c l c r} 1 & \times & 8 & = & 8\\ 11 & \times & 88 & = & 968\\ 111 & \times & 888 & = & 98568\\ 1111 & \times & 8888 & = & 9874568\\ 11111 & \times & 88888 & = & 987634568\\ 111111 & \times & 888888 & = & 98765234568\\ \end{array}

What is the value of 1111111 × 8888888 1111111 \times 8888888 , the next line in the series?


The answer is 9876541234568.

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1 solution

Terry Yu
Nov 3, 2017

The pattern is,

1 × 8 = 0 0 8 1 \times 8 = \color{#3D99F6}0 \color{#D61F06} 0 \color{#333333}8

11 × 88 = 9 6 8 11 \times 88 = \color{#3D99F6}9 \color{#D61F06} 6 \color{#333333}8

111 × 888 = 98 56 8 111 \times 888 = \color{#3D99F6}98 \color{#D61F06} 56 \color{#333333}8

1111 × 8888 = 987 456 8 1111 \times 8888 = \color{#3D99F6}987 \color{#D61F06} 456 \color{#333333}8

11111 × 88888 = 9876 3456 8 11111 \times 88888 = \color{#3D99F6}9876 \color{#D61F06} 3456 \color{#333333}8

111111 × 888888 = 98765 23456 8 111111 \times 888888 = \color{#3D99F6}98765 \color{#D61F06} 23456 \color{#333333}8

It also took me awhile to see this pattern, but blue is going downwards while red is going upwards, and black is staying at 8. Therefore, we found the pattern, so the next line in the series would be

1111111 × 8888888 = 987654 123456 8 1111111 \times 8888888 = \large\boxed{\color{#3D99F6}987654 \color{#D61F06} 123456 \color{#333333}8}

You can also verify that with a calculator.

Is there a proof for why it follows? I mean the pattern can break in the next number immediately

Mahdi Raza - 11 months, 2 weeks ago

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