Question

Which summation formula represents the series below? 1 + 2 + 6 + 24

Answer 1

Question:

Which summation formula represents the series below? 1 + 2 + 6 + 24

(a) $\sum_{n=2}^{5}(n-1) !$

(b) $\sum_{n=0}^{3} n !$

(c) $\sum_{n=1}^{4}(n+1) !$

(d) $\sum_{n=2}^{5} n !$

Answer:

Option a: $\sum_{n=2}^{5}(n-1) !$ is the correct answer.

Explanation:

Option a: $\sum_{n=2}^{5}(n-1) !$

By substituting the values of n and expanding the summation, we have,

$(2-1) !+(3-1) !+(4-1) !+(5-1) !$

Subtracting, we have,

$1 !+2!+3 !+4 !$

Expanding the factorial,

$1+(2*1)+(3*2*1)+(4*3*2*1)$

Simplifying, we get,

$1+2+6+24$

Thus, the summation $\sum_{n=2}^{5}(n-1) !$ represents the series $1+2+6+24$

Hence, Option a is the correct answer.

Option b: $\sum_{n=0}^{3} n !$

By substituting the values of n and expanding the summation, we have,

$0!+1!+2!+3!$

Expanding the factorial,

$0+1+(2*1)+(3*2*1)$

Simplifying, we get,

$0+1+2+6$

Thus, the summation $\sum_{n=0}^{3} n !$ does not represents the series $1+2+6+24$

Hence, Option b is not the correct answer.

Option c: $\sum_{n=1}^{4}(n+1) !$

By substituting the values of n and expanding the summation, we have,

$(1+1) !+(2+1) !+(3+1) !+(4+1) !$

Adding, we have,

$2!+3!+4!+5!$

Expanding the factorial,

$(2*1)+(3*2*1)+(4*3*2*1)+(5*4*3*2*1)$

Simplifying, we get,

$2+6+24+120$

Thus, the summation $\sum_{n=1}^{4}(n+1) !$ does not represents the series $1+2+6+24$

Hence, Option c is not the correct answer.

Option d: $\sum_{n=2}^{5} n !$

By substituting the values of n and expanding the summation, we have,

$2!+3!+4!+5!$

Expanding the factorial,

$(2*1)+(3*2*1)+(4*3*2*1)+(5*4*3*2*1)$

Simplifying, we get,

$2+6+24+120$

Thus, the summation $\sum_{n=2}^{5} n !$ does not represents the series $1+2+6+24$

Hence, Option d is not the correct answer.

Hence, the correct answer is Option a: $\sum_{n=2}^{5}(n-1) !$