Question

Study the pattern and write the next two steps. Can you say how the pattern works?

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1234 × 8 + 4 = 9876

12345 × 8 + 5 = 98765

Answer 1

According to the given pattern, the next two terms will be given by expression, $123456\times 8+6$ and $1234567\times 8+7$.

Solve $123456\times 8+6$ and $1234567\times 8+7$ to obtain the next two terms.

$123456\times 8+6=987654$

$1234567\times 8+7=9876543$

For $123456\times 8+6$, the pattern works by splitting $123456$ as

123456 = (111111 + 11111 + 1111 + 111 + 11 + 1)

123456 × 8 = (111111 + 11111 + 1111 + 111 + 11 + 1) × 8

= 111111 × 8 + 11111 × 8 + 1111 × 8 + 111 × 8 + 11 × 8 + 1 × 8

= 888888 + 88888 + 8888 + 888 + 88 + 8

= 987648

So, 123456 × 8 + 6 = 987648 + 6

= 987654.

Similarly for $1234567\times 8+7$ the pattern works by splitting $1234567$ as \

1234567 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1)

1234567 × 8 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) × 8

= 1111111 × 8 + 111111 × 8 + 11111 × 8 + 1111 × 8 + 111 × 8 + 11 × 8 + 1 × 8

= 8888888 + 888888 + 88888 + 8888 + 888 + 88 + 8

= 9876536

So, 1234567 × 8 + 7 = 9876536 + 7

= 9876543.