
Shortcut Trick
Identify the series as an Arithmetic Progression (AP) where the first term is 6 and the last term is 40.
Average of an AP = (First term + Last term) ÷ 2 = (6 + 40) ÷ 2 = 23.
Number of terms (n) = [(Last − First) ÷ Common Difference] + 1 = [(40 − 6) ÷ 2] + 1 = 18.
Sum of the series = Average × n = 23 × 18 = 414.
∴ The correct answer is 414.

Alternate Method
Given series: 6, 8, 10, 12, 14, ............., 40
First term (a) = 6
Last term (l) = 40
Common difference (d) = 8 − 6 = 2

Formula for the nth term of an AP: l = a + (n − 1)d
⇒ 40 = 6 + (n − 1) × 2
⇒ 40 − 6 = (n − 1) × 2
⇒ 34 = (n − 1) × 2
⇒ n − 1 = 17
⇒ n = 18
Formula for the Sum (Sn) of an AP: Sn = (n/2) × (a + l)
⇒ S18 = (18 ÷ 2) × (6 + 40)
⇒ S18 = 9 × 46
⇒ S18 = 414
∴ The correct answer is 414.

Additional Information
Arithmetic Progression (AP)
A sequence where the difference between consecutive terms is constant. nth term Tn = a + (n − 1)d.
Sum of First n Natural Numbers
The sum of the first n positive integers is calculated as [n × (n + 1)] ÷ 2.
Sum of First n Even Numbers
The sum of the first n even natural numbers starting from 2 is given by n × (n + 1).
General Sum Formula
For any AP, the sum is also Sn = (n/2) × [2a + (n − 1)d].