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The number pattern below shows the sum of consecutive odd numbers. Complete lines 5 and 6, find the last number in line 10, and the sum in line 12.

Mathematics

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Answers

3 parts
2.2

19\text{19}

2.3

144\text{144}

2.4

400\text{400}

Step-by-step solution

9 steps

Here are the solutions to the number pattern questions:

Part 2.1· 3 steps

Complete lines 5 and 6 in the number pattern below

  1. Identify the pattern.

    The pattern shows the sum of consecutive odd numbers starting from 1.
    Line nn sums the first nn odd numbers.
    The sum for Line nn is n2n^2.
    The last number in Line nn is the nn-th odd number, given by 2n−12n-1.

  2. Complete Line 5.

    Line 5 will sum the first 5 odd numbers. The 5th odd number is 2(5)−1=92(5)-1 = 9.
    The sum for Line 5 is 52=255^2 = 25.

    Line 5: 1+3+5+7+9=25=5×5\text{Line 5: } 1+3+5+7+9 = 25 = 5 \times 5
  3. Complete Line 6.

    Line 6 will sum the first 6 odd numbers. The 6th odd number is 2(6)−1=112(6)-1 = 11.
    The sum for Line 6 is 62=366^2 = 36.

    Line 6: 1+3+5+7+9+11=36=6×6\text{Line 6: } 1+3+5+7+9+11 = 36 = 6 \times 6

    The completed lines are:
    Line 5: 1+3+5+7+9=25=5×51+3+5+7+9 = 25 = 5 \times 5
    Line 6: 1+3+5+7+9+11=36=6×61+3+5+7+9+11 = 36 = 6 \times 6

Part 2.2

19\text{19}

In Line 4 the last number in the pattern is 7. What would be the last number in Line 10?

  1. Determine the formula for the last number.

    The last number in Line nn is the nn-th odd number, which is given by the formula 2n−12n-1.

  2. Calculate the last number for Line 10.

    For Line 10, n=10n=10.
    Last number =2(10)−1=20−1=19= 2(10)-1 = 20-1 = 19.
    The last number in Line 10 would be 19\boxed{\text{19}}.

Part 2.3

144\text{144}

What would all of the numbers in the pattern add up to in Line 12?

  1. Determine the formula for the sum.

    The sum of the numbers in Line nn is n2n^2.

  2. Calculate the sum for Line 12.

    For Line 12, n=12n=12.
    Sum =122=144= 12^2 = 144.
    All the numbers in Line 12 would add up to 144\boxed{\text{144}}.

Part 2.4

400\text{400}

Use what you have found above to describe a quick way of adding up the first 20 odd numbers.

  1. Apply the observed pattern.

    From the pattern, the sum of the first nn odd numbers is n2n^2.

  2. Describe the quick way.

    To add up the first 20 odd numbers, you simply square the number of odd numbers, which is 20.
    So, the sum is 202=40020^2 = 400.
    The quick way to add up the first 20 odd numbers is to square the number 20, which gives 400\boxed{\text{400}}.

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