The summation notation for the series 500+490+480+ . . . . . . . +20+10 is
[tex]25500-10\sum\limits^{50}_{n=1} {n}[/tex]
A summation notation is used to express a long summation into a single notation.
The given series is:
500+490+480+ . . . . . . . +20+10
This is an arithmetic series with:
a(1) = 500, and
d = 490 - 500 = -10
The last term is a(n) = 10. Find n first by using the nth term formula of an arithmetic sequence:
a(n) = a(1) + (n-1) . d
10 = 500 + (n-1) . (-10)
10 = 500 -10n + 10
10 n = 500
n = 50
Write the explicit formula for the nth term:
a(n) = a(1) + (n-1) . d
a(n) = 500 + (n-1) . (-10)
a(n) = 500 -10n + 10
a(n) = 510 - 10n
The series is the summation notation from n=1 to n = 50
[tex]=\sum\limits^{50}_{n=1} {a(n)}[/tex]
[tex]=\sum\limits^{50}_{n=1} {(510-10n)}[/tex]
[tex]=\sum\limits^{50}_{n=1} {510} -\sum\limits^{50}_{n=1} {10n}[/tex]
[tex]=510\times50-10\sum\limits^{50}_{n=1} {n}[/tex]
[tex]=25500-10\sum\limits^{50}_{n=1} {n}[/tex]
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