The length of substrings are **n** or **n-1** or **n-2** or....or **2** or **1** or **0**

let the string is **abcd...xyz**

no.of substrings of length **'n'** = 1 ------ abcd....xyz

no.of substrings of length **'n-1'** = 2 ------ abcd....xy , bcd....xyz

no.of substrings of length **'n-2'** = 3 ------ abcd....x , bcd....xy , cdef.....xyz

.......

no.of substrings of length **'2'** = n-1 ----- ab,bc,cd,....vx,xy,yz

no.of substrings of length **'1'** = n ------ a,b,c,d,....x,y,z

no.of substrings of length **'0'** = 1 ----> only empty string is possible

total substrings = 1+2+3+...+(n-1) + n + 1

= [ 1+2+3+...+(n-1) + n ] + 1

= $\frac{n(n+1)}{2}$ + 1

= **∑n **+ 1

NOTE :-

Trivial substrings :- empty string ( which is length = 0 ) and the original string ( which is length = n ) are trivial strings.

** ∴ No.of Trivial substrings are 2 for any non-empty string**

Non- Trivial substrings :- which are not Trivial substrings are called as Non- Trivial substrings.

** ∴ No.of Non-Trivial substrings are ∑n - 1**

Proper Substrings :- the substring which length is less than actual string

** ∴ No.of Non-Trivial substrings are ∑n**

Non-empty Substrings :- the substring which length is grater than 0

** ∴ No.of Non-Empty substrings are ∑n**