The sum of the given sequence is -24846.
Step-by-step explanation:
The given Arithmetic sequence is 46 + 42 +38... +(-446) +(-450).
- The first term of the sequence = 46
- The last term of the sequence = -450
- The common difference ⇒ 46 - 42 = 4
To find the number of terms in the sequence :
The formula used is [tex]n = (\frac{a_{n}-a_{1}} {d})+1[/tex]
where,
- n is the number of terms.
- [tex]a_{n}[/tex] is the late term which is -450.
- [tex]a_{1}[/tex] is the first term which is 46.
- d is the common difference which is 4.
Therefore, [tex]n =(\frac{-450-46}{4}) +1[/tex]
⇒ [tex]n = (\frac{-496}{4}) + 1[/tex]
⇒ [tex]n = -124 + 1[/tex]
⇒ [tex]n = -123[/tex]
⇒ n = 123, since n cannot be negative.
∴ The number of terms, n = 123.
To find the sum of the arithmetic progression :
The formula used is [tex]S = \frac{n}{2}(a_{1} + a_{n} )[/tex]
where,
- S is the sum of the sequence.
- [tex]a_{1}[/tex] is the first term which is 46.
- [tex]a_{n}[/tex] is the late term which is -450.
Therefore, [tex]S = \frac{123}{2}(46+ (-450))[/tex]
⇒ [tex]S = \frac{123}{2}(-404)[/tex]
⇒ [tex]S = 123 \times -202[/tex]
⇒ [tex]S = -24846[/tex]
∴ The sum of the given sequence is -24846.
