The sum of the given sequence is -6384.
Step-by-step explanation:
The given Arithmetic sequence is 14 + 8 + 2+ ... + ( 274) + (-280).
To find the number of terms in the sequence :
The formula used is [tex]n = (\frac{a_{n}-a_{1}} {d})+1[/tex]
where,
Therefore, [tex]n =(\frac{-280-14}{6}) +1[/tex]
⇒ [tex]n =( \frac{-294}{6}) + 1[/tex]
⇒ [tex]n = -49 + 1[/tex]
⇒ [tex]n = -48[/tex]
⇒ n = 48, since n cannot be negative.
∴ The number of terms, n = 48.
To find the sum of the arithmetic progression :
The formula used is [tex]S = \frac{n}{2}(a_{1} + a_{n} )[/tex]
where,
Therefore, [tex]S = \frac{48}{2}(14+ (-280))[/tex]
⇒ [tex]S = \frac{48}{2}(-266)[/tex]
⇒ [tex]S = 48 \times -133[/tex]
⇒ [tex]S = -6384[/tex]
∴ The sum of the given sequence is -6384.
