Consider the following sequence:
- Find the sum of 2 + 6 + 10 + 14 + ... + 54.
- A given line of the sequence is as follows: 2 + 6 + 10 + 14 + ... + i = 242 = 2 x j2. Find the value of j.
- Calculate the sum of Line 40 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
12Sum of 2 numbers:
2 + 6 = 8 =
2 x
22Sum 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 54 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 - 254 = Number of numbers x 4 - 2
54 + 2 = Number of numbers x 4
56 = Number of numbers x 4
Number of numbers
= 56 ÷ 4
= 14
Sum of 14 numbers
= 2 x Number of numbers in the string
2
= 2 x 14
2= 2 x 14 x 14
= 2 x 196
= 392
(b)
2 x j
2 = 242
j
2 = 242 ÷ 2
j
2 = 121
j = √121
j = 11
(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)
2Sum of Line
40 =
2 x (
40 + 1)
2 = 2 x 41
2 = 2 x 41 x 41
= 3362
Answer(s): (a) 392; (b) 11; (c) 3362