Question

Express the following numbers as the sum of odd numbers using the above pattern?
(a) 6³ (b) 8³ (c) 7³

Consider the following pattern.
2³-1³ =1+ 2 x 1 x 3
3³-2³ =1+ 3 x 2 x 3
4³-3³ =1+ 4 x 3 x 3
Using the above pattern, find the value of the following.
(i) 7³-6³ (ii) 12³-11³ (iii)20³-19³ (iv) 51³-50³ 

Answer 1

(a) n=6 and (n-1) = 5
∴ We start with (6 x 5) + 1 = 31
we have
6³= 31 + 33 + 35 + 37 + 39 + 41=216
(b) n=8 and (n-1)=7
∴ we start with (8 x 7)+1=57
we have
8³= 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512
(c) n = 7 and (n-1) =6
∴ we start with (7 x 6) + 1 = 43
We have
7³= 43+ 45+ 47 + 49 + 51 + 53 + 55 = 343

(i) 7³-6³ = 1 + 7 x 6 x 3 = 127
(ii) 12³-11³ = 1 + 12 x 11 x 3 = 397
(iii) 20³-19³ = 1 + 20 x 19 x 3 = 1141
(iv) 51³-50³ = 1 + 51 x 50 x 3 = 7651