Test: Introduction To Sequences

a_n = (n²)/2ⁿ
⇒ a₅ = [(5)²]/2⁽⁵⁾
⇒ a₅ = 25/32

The correct option is A.

A sequence is an enumerated collection of objects in which repetitions are allowed and order does matter. Like a set, it contains members. The number of elements is called the length of the sequence. A sequence is a function whose domain is the set of natural numbers or a subset of the natural numbers.

a1 = 2 
a2 = 2a1 + 1
=> 2(2) + 1 = 5
a3 = 2a2 + 1
=> 2(5) + 1 = 11
a4 = 2a3 + 1
=> 2(11) + 1 = 23
Hence, the required series is : 2,5,11,23………

an = (n-1)(2-n)(3+n)
Put n = 10
an = 9×(-8)×13
= - 936

an = 2(n -1)(2n - 1) 
a10 = 2(10-1)(2(10)-1))
= 2(9)(19)
= 342

Put n=1 then a₁=1/2  

then put n=2 a₂=3/2

 put n=3 a₃=5/2

 n=4 a₄=7/2

 n=5 a₅=9/2

 their sum is 25/2

How do we get from 2 to 6? One way is to multiply by 3.

How do we get from 6 to 18? We can multiply by 3 once again.

What about 18 to 54? Again, we can multiply by 3.

We notice that our common ratio is 3. We can leverage this fact to write the next terms of our sequence:

...54,(54⋅3),(54⋅32),(54⋅33)

Notice, we are multiplying by three every time. The 7th term of this sequence is given by the blue expression

54⋅33, which is equal to

54⋅27=1458

Those sequences whose terms follow certain patterns are called progressions

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant.