Question

Solve the following recurrence relation through substitution. (i) an = an-1 + 5, subject to initial condition a1 = 2

Answer 1

an = a(n -1) + 5

we know,
an = a1 + ( n -1)d
and
a(n -1) = a1 + ( n -1-1)d = a1 + ( n -2)d

where a1 and d are first term and common difference of an AP

now,

a1+ ( n -1)d = a1 + ( n -2)d + 5

d = 5
hence,

an = a1+ ( n - 2) 5 + 5
but a1= 2

so,
an = 2 + 5n -10 + 5 = -3 + 5n

an = 5n -3