Question

What are the next two numbers in this sequence: \[ - 2,6, - 24,120, - 720,5040,...?\]

Answer 1

Hint: An infinite sequence is a list or string of discrete objects, usually numbers that can be paired off one-to-one with the set of integers. There is generally some relationship between numbers in infinite sequence.

Complete step-by-step answer:
A sequence is an ordered list of objects (or events). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters in a sequence, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable, totally ordered set, such as the natural numbers.
Examples: \[\left( {C,A,T} \right)\] is a different sequence from \[\left( {T,A,C} \right)\] . Also, the sequence \[\left( {1,2,3,3,5} \right)\] which contains the number \[3\] at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers \[\left( {1,2,3,4,5,6,...} \right)\] .
An Arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
In the given sequence, we can observe that each term is divisible by previous terms and the quotient keeps on increasing. It is explained as follows:
 \[\dfrac{6}{{ - 2}} = - 3\] , \[\dfrac{{ - 24}}{6} = - 4\] , \[\dfrac{{120}}{{ - 24}} = - 5\] , \[\dfrac{{ - 720}}{{120}} = - 6\] and \[\dfrac{{5040}}{{ - 720}} = - 7\] .
We can see that the quotients are negative and in increasing order forming an arithmetic series i.e. \[ - 3, - 4, - 5,...\]
Hence the next term can be found out as follows by taking missing variable as \[x\] and \[y\] :
 \[\dfrac{x}{{5040}} = - 8\]
 \[x = 5040( - 8) = - 40320\]
Next term will be as follows:
 \[\dfrac{y}{{ - 40320}} = - 9\]
 \[y = ( - 40320)( - 9) = 362880\]
Hence the next two terms in the series will be \[ - 40320\] and \[362880\] .
So, the correct answer is “ \[ - 40320\] and \[362880\] ”.

Note: here are 4 types of sequence. Arithmetic, Geometric, Harmonic and Fibonacci numbers. The given sum is based on arithmetic sequence since the difference between the consecutive quotients is equal i.e. series \[ - 3, - 4, - 5, - 6,...\] has constant consecutive difference:
 \[( - 4) - ( - 3) = - 1\]
 \[( - 5) - ( - 4) = - 1\]
 \[( - 6) - ( - 5) = - 1\]
The above sum can also be solved by analysing the relationship between the numbers in sequence using the multiplication method. E.g.
 \[
  6 = - 2 \times 3 \\
   - 24 = 6 \times - 4 \\
  120 = - 24 \times 5 \\
   - 720 = 120 \times - 6 \;
 \]
We can see that there is a successive increase in multiplication digit for next term and alternative numbers are negative.