Consider the following sequence:

1. Find the sum of 2 + 6 + 10 + 14 + ... + 62.
2. A given line of the sequence is as follows: 2 + 6 + 10 + 14 + ... + m = 648 = 2 x n². Find the value of n.
3. Calculate the sum of Line 36 if the pattern for the sum of the lines is as follows:-
   Line 1: 8
   Line 2: 8 + 10
   Line 3: 8 + 10 + 12

(a)
Sum of 1 number: 2 = 2 x 1²
Sum of 2 numbers: 2 + 6 = 8 = 2 x 2²
Sum of 3 numbers: 2 + 6 + 10 = 18 = 2 x 3² 
Sum of 4 numbers: 2 + 6 + 10 + 14 = 32 = 2 x 4²

Formula:
Sum of numbers = 2 x Number of numbers in the string²

To find out how many numbers are in the string if 62 is the last number.

Last number in sum of 1 number: 2 = 1 x 4 - 2
Last number in sum of 2 numbers: 6 = 2 x 4 - 2
Last number in sum of 3 numbers: 10 = 3 x 4 - 2
Last number in sum of 4 numbers: 14 = 4 x 4 - 2

Formula:
Last number = Number of numbers in the string x 4 - 2

62 = Number of numbers x 4 - 2
62 + 2 = Number of numbers x 4
64 = Number of numbers  x 4

Number of numbers
= 64 ÷ 4
= 16

Sum of 16 numbers
= 2 x Number of numbers in the string²
= 2 x 16²
= 2 x 16 x 16
= 2 x 256
= 512

(b)
2 x n² = 648
n² = 648 ÷ 2
n² = 324
n = √324
n = 18

(c)
Pattern for the sum of the lines
Line 1: 8 = 2 x (1 + 1)²
Line 2: 8 + 10 = 18 = 2 x (2 + 1)² 
Line 3: 8 + 10 + 12 = 32 = 2 x (3 + 1)² 

Formula:
Sum of Line Number = 2 x (Line Number + 1)²

Sum of Line 36
= 2 x (36 + 1)²
= 2 x 37²
= 2 x 37 x 37
= 2738

Answer(s): (a) 512; (b) 18; (c) 2738