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

Answer:

$\sum\limits_{i=1}^{n} c \cdot a_i \ is \ equal \ to \ c = \ c \cdot \sum\limits_{i=1}^{n} a_i$

$\sum\limits_{i=1}^{n} i \ is \ equal \ to \ b = \dfrac{n \cdot (n + 1)}{2}$

$\sum\limits_{i=1}^{n} c \ is \ equal \ to \ d = c \cdot n$

$\sum\limits_{i=1}^{n} i^3 \ is \ equal \ to \ e = \left [ \dfrac{n \cdot (n + 1)}{2} \right]^2$

$\sum\limits_{i=1}^{n} i^2 \ is \ equal \ to \ a = \dfrac{n \cdot (n + 1) \cdot (2 \cdot n + 1)}{6}$

Step-by-step 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;

$\sum\limits_{i=1}^{n} c \cdot a_i= c \cdot \sum\limits_{i=1}^{n} a_i = c$

$\sum\limits_{i=1}^{n} i = \dfrac{n \cdot (n + 1)}{2} = b$

$\sum\limits_{i=1}^{n} c = c \cdot n = d$

$\sum\limits_{i=1}^{n} i^3 = \left [ \dfrac{n \cdot (n + 1)}{2} \right]^2 = e$

$\sum\limits_{i=1}^{n} i^2 = \dfrac{n \cdot (n + 1) \cdot (2 \cdot n + 1)}{6} = a$.