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What are the last number in line 10 and the sum of numbers in line 12, and how do you quickly add the first 20 odd numbers?

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

6 steps

Here are the solutions to the questions:

Part 2.2· 2 steps

  1. Identify the pattern for the last number in each line.

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

  2. Calculate the last number for Line 10.

    For Line 10, substitute n=10n=10 into the formula:

    2(10)−1=20−1=192(10) - 1 = 20 - 1 = 19

    The last number in Line 10 is 19\boxed{\text{19}}.

Part 2.3

144\text{144}

  1. Identify the pattern for the sum of numbers in each line.

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

  2. Calculate the sum for Line 12.

    For Line 12, substitute n=12n=12 into the formula:

    122=14412^2 = 144

    All of the numbers in Line 12 add up to 144\boxed{\text{144}}.

Part 2.4

400\text{400}

To quickly add up the first 20 odd numbers, you can use the pattern observed: the sum of the first nn odd numbers is equal to n2n^2.

  1. Identify the number of odd numbers to be added.

    We need to add the first 20 odd numbers, so n=20n=20.

  2. Apply the pattern to find the sum.

    The sum of the first 20 odd numbers is 20220^2.

    202=40020^2 = 400

    A quick way to add up the first 20 odd numbers is to square the number of odd numbers, which is 202=40020^2 = \boxed{\text{400}}.

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