Question

use the method of undetermined coefficient to find the sum of 1+2+3+.......+(n-1)+n

Answer 1

Answer:

$1 + 2 + 3 + ....... + n  = \frac{n(n + 1) }{2}$

Explanation :

Let's start by assuming that the above summation can be expressed as a polynomial function f(n) of the form ;

f(n) = A₀ + A₁n + A₂n² + A₃n³ + .....

Let's equate this function to our original summation:

1+2+3+.......+(n-1)+n = A₀ + A₁n + A₂n² + A₃n³ + .....        -----------------------(1)

We know that,

f(n+1) =  1+2+3+.......+n + (n+1)  =  A₀ + A₁(n+1) + A₂(n+1)² + A₃(n+1)³ + ..... ------(2)

Equation (2) -Equation (1)

n + 1 = A₁ + (2n + 1)A₂ + (3n²+3n+1)A₃ + .....

we can eliminate all variables on the right except for n + 1 = A₁ + (2n + 1)A₂ because all other values are 0. Then the new equation is

n + 1 = A₁ + (2n + 1)A₂

n + 1 = A₁ + A₂ + 2A₂n

2A₂ = 1   and A₁ + A₂ = 1

A₁ = A₂ = $\frac{1}{2}$  put in equation 1

1 + 2 + 3 + ..... + n = A₀ + n/2 + n²/2 + 0 + 0 ......

                            =  A₀ + n/2 + n²/2

Now we need to find the value of A₀

Suppose we put n = 1

We get ,

1 = A₀ + 1

A₀ = 0

Therefore, we have now completely simplified the equation. Expressing it in proper form, we have

1 + 2 + 3 + ....... + n = $0 + \frac{n}{2}  + \frac{n^{2} }{2}$

1 + 2 + 3 + ....... + n = $\frac{n^{2} + n }{2}$

$1 + 2 + 3 + ....... + n  = \frac{n(n + 1) }{2}$

Answer 2

Answer:

is the answer correct for this question..

Step-by-step explanation:

if correct text me please..