Abstraction Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
A student notices: 1+3=4=221+3=4=2^2, 1+3+5=9=321+3+5=9=3^2, 1+3+5+7=16=421+3+5+7=16=4^2. State the general pattern as an abstraction and verify it for n=5n=5.

Solution

  1. 1
    The pattern: the sum of the first nn odd numbers equals n2n^2. In symbols: ∑k=1n(2k−1)=n2\sum_{k=1}^{n}(2k-1) = n^2.
  2. 2
    Verify for n=5n=5: 1+3+5+7+9=25=521+3+5+7+9 = 25 = 5^2. Correct.

Answer

∑k=1n(2k−1)=n2\sum_{k=1}^{n}(2k-1) = n^2
Recognising a pattern in specific cases and writing it as a general formula is the essence of abstraction. Verifying the formula in new cases gives confidence before attempting a proof.

About Abstraction

The cognitive and mathematical process of identifying essential features shared by many specific cases and ignoring irrelevant details.

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More Abstraction Examples

Example 1 easy

The rule '[formula]' applies to a [formula] room, a [formula] phone screen, and a [formula] field. E

Example 2 medium

The statement '[formula] for all real numbers [formula]' abstracts what concrete arithmetic observat

Example 3 easy

Identify what is being abstracted: 'The [formula]-th term of an arithmetic sequence is [formula].'

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