Sequences and Series Test - 1

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

The 7th term of a geometric progression (GP) can be found using the formula:

Tn=a ⋅r^{n −1}

Where:

• Tn ​is the n-th term,
• a  is the first term,
• r  is the common ratio,
• n  is the term number.

In the given GP: 2, 6, 18, ...

• The first term a=2
• The common ratio r=6/2=3

Now, substitute into the formula for the 7th term:

T₇   = 2 ⋅3^{7 −1}   = 2 ⋅3⁶   = 2 ⋅729 = 1458

So, the 7th term is 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.