Respuesta :
Answer:
[tex]S_{n} = 36n[/tex]
This shows that the sum of the series varies directly as the number of terms.
Step-by-step explanation:
The terms of the airthmetic series is given by .
[tex]a_{n}=a_{1}+(n-1)d[/tex]
Where [tex]a_{1}[/tex] is the first term .
n represented the number of terms in the airthmetic series .
d is the common difference .
As given
If the first and the last terms of an arithmetic series are 10 and 62 .
[tex]a_{1}= 10[/tex]
[tex]a_{n}= 62[/tex]
Putting in the above
[tex]62=10+(n-1)d[/tex]
62 - 10 = (n-1)d
52 = (n-1)d
[tex]d = \frac{52}{(n-1)}[/tex]
The Sum of the nth terms of the airthmetic series is given by .
[tex]S_{n} = \frac{n}{2}(2a_{1}+(n-1)d)[/tex]
Putting the values in the above
[tex]S_{n} = \frac{n}{2}(2\times 10+(n-1)\frac{52}{(n-1)})[/tex]
[tex]S_{n} = \frac{n}{2}(2\times 10+52)[/tex]
[tex]S_{n} = \frac{n}{2}(20+52)[/tex]
[tex]S_{n} = \frac{n}{2}(72)[/tex]
[tex]S_{n} = \frac{72n}{2}[/tex]
[tex]S_{n} = 36n[/tex]
(This is for the n terms )
Therefore this shows that the sum of the series varies directly as the number of terms.
