DHARM

DISCRETE NUMERIC FUNCTIONS AND GENERATING FUNCTIONS 31

Example 2.8. Let an be a numeric function s.t.

an = RS1for0n503 for n 51≤≤≥
T
Find S–25 an.
Sol. Using definition we have S– 25 an = an+25 for n ≥ 0. To determine the numeric function
an+25, for n ≥ 0, we have

• Since an = 1 for 0 ≤ n ≤ 50, also an+25 = 1 for 0 ≤ n + 25 ≤ 50 or 0 ≤ n ≤ 25.


• For n ≥ 51, an = 3, also an+25 = 3 for n + 25 ≥ 51 or n ≥ 26.

Therefore, S–25 an = SR^1025226 forfor ≤≤≥

T

n

n

2.2.7. Let an be a numeric function then accumulated sum of an is defined by

an

k

∑ (for k = 0 to n)

is a numeric function. For example if an describes the monthly income of worker then its

annual income is given by the accumulated sum (bn) of an where,

bn =

k

an

=

∑

0

12

Example 2.9. Let an = P (1.11)n for n = 0. Find accumulated sum of an for n = m.

Sol. Let bn be the accumulated sum of an, i.e.,

bn =

k

m

an

=

∑

0

=

k

m

k

=

∑

0

P(. ) (^111) for m ≥ 0
= P (1 + 1.11 + 1.11^2 + ........+ 1.11m)
= P (1.11m+1 – 1)/ (1.11 – 1)
= 100 P (1.11m+1 – 1)/11 for m ≥ 0
2.2.8 Let an be a numeric function then its forward difference is denoted by ∆an, and
defined as,
∆an = an+1 – an for n = 0
Likewise, backward difference is denoted by ∇an, and defined as,
∇=

−≥
R
S
T −
a
an
n aann n
0
1
0
1
for
for
For example, if an describes the annual income of a worker then,

• ∆an will describe the change in income from nth year to (n + 1)th year, and


• ∇an will describe the change in income from nth year over (n – 1)th year.
  Consider a numeric function an. i.e.

a

n

n= n

≤≤

≥

R

S

T

07

18

for 0

for

then ∆an = an+1 – an for n ≥ 0