Quantitative Methods I
Academic Year 23-24
EXERCISES FOR CHAPTER 11
SUMMATIONS
1- Expand the following sums:
a) ∑ f)
b) ∑ c) ∑
∑
g) ∑ 1 h) ∑ 2
d) ∑
i)
∑ j)
∑
a) ∑ i)
∑ $5 √4 '
b) ∑ j)
∑** () ∑
() 1
e) ∑ 2√ 2
2- Compute the following summations:
c) ∑ 2
k) ∑, +,
d) ∑ e) ∑
f)
∑
!
,-
∑ 1 m) ∑ $5 3 '
g) ∑# 2
h) ∑
l)
n) ∑ ) ∙ 2
o) ∑, 2 1
"
p) ∑ $5 ∙ 3
'
3- Express the following sums in summation notation:
a) 5 6 7 ⋯ 11
b) 3 9 27 81 243 c) 4 8 12 ⋯ 45
1
Based on exercises from book Essential Mathematics for Economic Analysis, Sydsaeter & Hammond.
1
Quantitative Methods I
Academic Year 23-24
d) 1 8 27 ⋯ 5
e) 1 … 1
4- Insert the appropriate limits of summation in the right-hand side of the following sums:
a) ∑* 2) 3 7 ∑?? 2 1
?
,
b) ∑
29 7 ∑,? ∙ 9
c) ∑:
7 ∑? ? 32 ∙ 2
2
5- The arithmetic mean of T elements: , , , … , < , is defined as the sum of all the elements
divided by the number of elements and it is called => :
∑< ?
=> 7
Prove that:
a) ∑< => 7 0
b) ∑< => 7 ∑< ?=>
PRICE INDICES (Economic application)
6- In order to summarize the overall effect of price changes for several different goods within
a country, a number of alternative price indices have been suggested:
AB_DEFG HGE 7
AB_BEKℎG 7
∑ FI J
∑ F J
∑ FI JI
∑ F JI
2
∙ 100
∙ 100
Quantitative Methods I
Academic Year 23-24
Compute the Laspeyres and Paasche price index for a basket of 3 commodities (A, B y C)
where F and FI are the price per unit of good in year 0 and in year t, respectively.
a)
b)
F (price in base year)
FI (price in year t)
J (quantity in base year)
JI (quantity in year t)
A
1
2
4
3
B
3
2
4
5
C
8
4
7
7
F (price in base year)
FI (price in year t)
J (quantity in base year)
JI (quantity in year t)
A
2
3
3
4
RULES FOR SUMS AND FORMULAS
7- Apply the properties and summation formulas to compute the following sums:
a) ∑
M
e) ∑ 3 2
b) ∑
)
f)
g) ∑,
2 1
c) ∑
)
d)
∑
)
∑ 2 1
h) ∑
2 1
DOUBLE SUMS
8- Evaluate the following sums:
a) ∑ ∑ ∙ 3
b) ∑O ∑N +
NO
NO
h) ∑,
∑ 6 6) i)
∑,
∑ 2 1
c) ∑,
∑ j)
∑,
∑ 2 2)
d) ∑
∑ )
k) ∑,
∑ 3 3) e) ∑
∑ l)
f)
∑
∑ 1
∑ ∑, )
m) ∑,
∑ ) g) ∑
∑ n) ∑ ∑,
) 2 3
B
4
6
5
4
C
6
9
7
7
Quantitative Methods I
Academic Year 23-24
NEWTON’S BINOMIAL FORMULA (Application)
9- Calculate the following factorial expressions:
a) 0!
e) 25!
b) 1!
f)
! !
g)
!
c)
!
!
d) 25!
h)
!
! !
!
!
10- Calculate the following combinatorial expressions:
a) $ '
f)
$'
,
b) $'
g) $,'
c) $ '
h) $ '
,
d) $ '
i)
$,, '
j)
$'
a) f)
2 2 b) g) 1 2F
c) h) 2R 2
d) 2 i)
1 3H
e) 1 2Q
j)
2S 25
e) $ '
11- Use Newton’s binomial formula to find:
12- Find an element of the expansion:
a) 5th element of 2 b) 17th element of 22 10
4