let an be the sum of the squares of first n natural numbers. that is an = +...+ n^2 for n> or equal to 1.

Question

A). Find the values a1,a2,a3,a4,a5,a6, and a7.

 

B). Based on the computational results in a), deduce a formula for an

Answer 1

       n

a_{n =} ∑ n²

       1

 

   a₁₌ 1²  =1

 

    a₂= 1² + 2² =5

 

    a₃ = 1² + 2² + 3^{2 =} 14

 

 

Formula for Sum of the square of natural numbers

 

n( n+1) ( 2n +1) / 6

 

  that can be proven by mathematical induction

 

 For our data if we calculate for n=3

 

  3( 4) ( 7) /6  =14 

 

    a_{4 =} a _{3 +} 4² = 30

 

    a₅ = a₄  + 5^{2 = 55}

 

    a_{6  =} a_{5  + 6}2 = 81

 

    a _{7 =} a_{6 + 7}² = 130