Respuesta :
The sequence for first 41 positive odd numbers is as,
1, 3, 5, 7, 9, ....
This is a arithmetic sequence with first term as a = 1, common difference of d = 2 and number of terms with n = 41.
The formula for the sum of terms in arithmetic sequence is,
[tex]S=\frac{n}{2}\lbrack2a+(n-1)d\rbrack[/tex]Substitute the values in the formula to determine the sum of first 41 positive odd numbers.
[tex]\begin{gathered} S=\frac{41}{2}\lbrack2\cdot1+(41-1)\cdot2\rbrack \\ =\frac{41}{2}\lbrack2+40\cdot2\rbrack \\ =\frac{41}{2}\cdot82 \\ =41\cdot41 \\ =(41)^2 \end{gathered}[/tex]So sum of first 41 positive odd numbers is equal to square of 41. Thus a conjecture can be made that,
The sum is equal to the number of terms squared.
