Answer:
Sum of the arithmetic series of odd numbers 1 + 3 + 5 + ... + 25 is 169 .
Step-by-step explanation:
Formula of airthmetic series
[tex]a_{n}=a_{1}+(n-1)d[/tex]
[tex]S_{n} = \frac{n(a_{1}+a_{n})}{2}[/tex]
Where n is the nth term , [tex]a_{n}[/tex] is the nth term , d is the common difference and [tex]a_{1}[/tex] is the first term .
As the airthmetic series given in the question .
1 + 3 + 5 + ... + 25
[tex]a_{1}= 1[/tex]
[tex]a_{2}= 3[/tex]
[tex]d=a_{2}-a_{1}[/tex]
d = 3 - 1
d = 2
[tex]a_{n}= 25[/tex]
Put all the values in the formula
[tex]25=1+(n-1)2[/tex]
25 - 1 = 2n - 2
24 + 2 = 2n
26 = 2n
[tex]n = \frac{26}{2}[/tex]
n = 13
Now put the values in the another formula
[tex]S_{n} = \frac{13\times (1+25)}{2}[/tex]
[tex]S_{n} = \frac{13\times (26)}{2}[/tex]
[tex]S_{n} = 13\times 13[/tex]
[tex]S_{n} = 169[/tex]
Therefore the sum of the arithmetic series of odd numbers 1 + 3 + 5 + ... + 25 is 169 .
