Question

Enter the correct letter to match each summation expression with the property or formula. N Σ i=1 cai is equal to . A n(n + 1)(2n + 1) 6 n Σ i=1 i is equal to . B n(n + 1) 2 n Σ i=1 c is equal to . C c n Σ i=1 ai n Σ i=1 i3 is equal to . D cn n Σ i=1 i2 is equal to

Answer 1

c, b, d, e, a

Explanation:

Answer 2

`\sum\limits_(i=1)^(n) c \cdot a_i \ is \ equal \ to \ c = \ c \cdot \sum\limits_(i=1)^(n) a_i`

`\sum\limits_(i=1)^(n) i \ is \ equal \ to \ b = (n \cdot (n + 1))/(2)`

`\sum\limits_(i=1)^(n) c \ is \ equal \ to \ d = c \cdot n`

`\sum\limits_(i=1)^(n) i^3 \ is \ equal \ to \ e = \left [ (n \cdot (n + 1))/(2) \right]^2`

`\sum\limits_(i=1)^(n) i^2 \ is \ equal \ to \ a = (n \cdot (n + 1) \cdot (2 \cdot n + 1))/(6)`

Explanation:

1) The sum of a series of the cubes of numbers 'i' is given as follows;

Sₙ = n/2 × (1st term + Last term)

∴ Sₙ = n/2 × (1 + n) = n·(n + 1)/2

It can be shown that the sum of a series of the cubes of numbers 'i²' is given as follows;

Sₙ = (n·(n + 1)·(2·n + 1))/6

It can be also be shown that the sum of a series of the cubes of numbers 'i³' is given as follows;

Sₙ = (n·(n + 1)/2)²

Therefore, we have;

`\sum\limits_(i=1)^(n) c \cdot a_i= c \cdot \sum\limits_(i=1)^(n) a_i = c`

`\sum\limits_(i=1)^(n) i = (n \cdot (n + 1))/(2) = b`

`\sum\limits_(i=1)^(n) c = c \cdot n = d`

`\sum\limits_(i=1)^(n) i^3 = \left [ (n \cdot (n + 1))/(2) \right]^2 = e`

`\sum\limits_(i=1)^(n) i^2 = (n \cdot (n + 1) \cdot (2 \cdot n + 1))/(6) = a`.