Specialization Math Example 3

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Example 3

easy
The general formula for the sum of a geometric series is Sn=a(rn−1)r−1S_n = \frac{a(r^n-1)}{r-1}. Specialise to a=1,r=2a=1, r=2 and compute S5S_5.

Solution

  1. 1
    Substitute a=1a=1, r=2r=2, n=5n=5: S5=1(25−1)2−1=311=31S_5 = \frac{1(2^5-1)}{2-1} = \frac{31}{1} = 31.
  2. 2
    Verify: 1+2+4+8+16=311+2+4+8+16 = 31. Confirmed.

Answer

S5=31S_5 = 31
Specialisation applies a general formula to a specific case. Verifying the result by direct computation builds confidence in both the formula and the specialisation.

About Specialization

Applying a general theorem or formula to a specific case by substituting particular values for the variables or parameters.

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More Specialization Examples

Example 1 easy

The Binomial Theorem states [formula]. Specialise to [formula] and [formula] to obtain two identitie

Example 2 medium

The AM-GM inequality states: for positive reals [formula], [formula]. Specialise to [formula] and [f

Example 4 medium

The general derivative rule is [formula]. Specialise to find the derivatives of [formula], [formula]

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GeneralizationEdge Cases