Structure Recognition Math Example 4

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

Example 4

medium
Recognise the structure in the sum 1+2+4+8+161 + 2 + 4 + 8 + 16 and compute it using the geometric series formula.

Solution

  1. 1
    Recognise: this is a geometric series with first term a=1a=1, common ratio r=2r=2, and n=5n=5 terms.
  2. 2
    Apply the formula: Sn=a(rn−1)r−1=1(25−1)2−1=32−11=31S_n = \dfrac{a(r^n-1)}{r-1} = \dfrac{1(2^5-1)}{2-1} = \dfrac{32-1}{1} = 31.
  3. 3
    Verify by direct addition: 1+2+4+8+16=311+2+4+8+16 = 31. Confirmed.

Answer

1+2+4+8+16=311+2+4+8+16 = 31
Once a sum is recognised as geometric, the closed-form formula gives the answer without adding each term. This is the power of structure recognition.

About Structure Recognition

The skill of identifying that a given mathematical expression or problem belongs to a known family or matches a recognizable pattern.

Learn more about Structure Recognition →

More Structure Recognition Examples

Example 1 easy

Recognise and name the algebraic structure in [formula], then factorise.

Example 2 medium

Identify the structure in [formula] and use it to simplify [formula].

Example 3 easy

Identify the structure and factorise: [formula].

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