Question

Determine the sum of the terms in the series 25 + 30 + 35 + ⋯ + 245. Show your work.

Answer 1

6075

Explanation:

• Given series is: 25 + 30 + 35 + ⋯ + 245.

• Here, first term (a) = 25
• Common difference (d) = 5
• Last term
  `(a_n) = 245`

• First we find the number of terms that this series has by the following formula:

• 
  `a_n = a+(n-1)d`

• Plugging the values of a, d and
  `a_n`, we find.

• 
  `245 = 25+(n-1)5`

• 
  `\implies 245 - 25=(n-1)5`

• 
  `\implies 220=(n-1)5`

• 
  `\implies (220)/(5)=n-1`

• 
  `\implies 44=n-1`

• 
  `\implies 44+1=n`

• 
  `\implies\red{\bold{ n=45}}`

• Thus, the given series contains 45 terms.

• Next, we find the sum of the given series using the formula given below.

• 
  `S_n = (n)/(2)(a+a_n)`

• 
  `\implies S_(45)= (45)/(2)(25+245)`

• 
  `\implies S_(45)= (45)/(2)(270)`

• 
  `\implies S_(45)=45*135`

• 
  `\implies\orange{\boxed{\boxed{\boxed{\bold{ S_(45)=6075}}}}}`