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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Sigma Notation

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:sigma_notation/4751/


Example 1.

Calculate .

    
Example 2.

Calculate .

    
3.

Calculate .

    
4.

With mathematical induction, show that:


    
5.

With mathematical induction, show that:


    
6.

Using mathematical induction, show that:


    
7.

With mathematical induction, prove that the sum of the first n terms of the geometric sequence is equal to:


    
8.

Prove by induction that the sum of the first n terms of the arithmetic sequence is equal to:


    

9.

Calculate , find the formula for the sum of the given finite terms and prove that the formula is general by mathematical induction:


    
10.

Calculate , find the formula for the sum of the given finite terms and prove the correctness of the formula by mathematical induction:


    
11.

Calculate . Find the formula for the sum of the given series and prove the correctness and generality of the formula by complete induction:


    
12.

Calculate . Find the formula for the sum of the given finite series and prove the correctness and generality of the formula by mathematical induction:


    
13.

With mathematical induction prove:


, for each

   
14.

Prove by induction that the sum of the interior angles in a convex n-gon is equal to:


    
15.

Prove:


  • that the sum of four powers of 5, whose exponents are consecutive natural numbers, is divisible by 26.


  • by mathematical induction that for every natural number n.

    
16.

Calculate:


  • The sum of the first 100 natural numbers.


  • The sum: .