What is the average of first twenty numbers in an Arithmetic Progression having common difference of 2.5 and first number of series is 10?
Given,
⇒ Average = Sum of terms/Number of terms
For Arithmetic Progression,
If Sum of terms = Sn and first term of progression = a, common difference = d and n = number of terms-
$(\Rightarrow {S_n} = \frac{n}{2}\left( {2a + (n - 1} \right)d))$
For given series -
a = 10, d = 2.5 and n = 20
$(\Rightarrow {S_{20}} = \frac{{20}}{2}\left\{ {2 \times 10 + \left( {20 - 1} \right) \times 2.5} \right\})$
⇒ S20 = 10{20 + 19 × 2.5}
⇒ S20 = 10(20 + 47.5)
⇒ S20 = 10 × 67.5
⇒ S20 = 675
Average of first 20 terms of Arithmetic Progression = 675/20 = 33.75
∴ Average of first 20 terms of Arithmetic Progression = 33.75