Problem 1 :
A GP has a common ratio of 0.65. Its sum to infinity is 120. What is the first term?
Solution :
Common ratio r = 0.65 if r < 1
S∞ = 120
Sn
= a/(1 – r)
S∞ = a/(1 – 0.65)
120 = a/0.35
a = 120 × 0.35
a = 42
So, the first term is 42.
Problem 2 :
Another GP has 2.8 as its first term and its sum to infinity is 3.2. Find its common ratio.
Solution :
First term a = 2.8
s∞ = 3.2
To find common ratio :
Sn = a/(1 – r)
S∞ = 2.8/(1 – r)
3.2 = 2.8/(1 – r)
Multiplying (1 – r) on both sides.
3.2(1 – r) = 2.8/(1 – r) × (1 – r)
3.2 – 3.2r = 2.8
Subtracting 3.2 on both sides.
3.2 – 3.2 – 3.2r = 2.8 – 3.2
-3.2r = -0.4
Dividing -3.2 on both sides.
-3.2r/-3.2 = -0.4/-3.2
r = 0.125
Problem 3 :
Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32 ,8, 2, 1/2.
Solution :
Product of corresponding terms = 2 × 128 + 4 × 32 + 8 × 8 + 16 × 2 + 32 × 1/2
= 256 + 128 + 64 + 32 + 16
= 64[4 + 2 + 1 + 1/2 + 1/22]
4, 2, 1, 1/2, 1/22 is a G.P.
(Sn) = a(1 – rn)/(1 – r)
a = 4
n = 5
r = 1/2 r < 1
(S5) = 4((1 - 1/2)5)/(1 – 1/2)
= 4(1 – 1/32)/1/2
= 8(32 – 1)/32
= 8(31)/32
= 31/4
= 64[31/4]
= 16 × 31
= 496
Problem 4 :
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Solution :
The number of ancestors are
2, 4, 8, 16, …
It is a G.P. Since,
4/2 = 2
8/4 = 2
So, common ratio is 2.
a = 2
Number of generations (n) = 10
We have to find the number of his ancestors during the ten generations preceding his own,
r = 2 if r > 1
Sum of n terms in GP (Sn) = a(rn – 1)/r – 1
S10 = 2(210 – 1)/2 – 1
S10 = (210 + 1 – 2)
= 211 – 2
= 2048 – 2
= 2046
Hence, the number of ancestors preceding the person is 2046.
Problem 5 :
We know the sum of the interior angles of a triangle is 180º. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, … sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
Solution :
Sum of interior angles of polygon = (n - 2) 180
The sum of the interior angles of a polygon with 3 sides :
180º
The sum of the interior angles of a polygon with 4 sides :
= (4 - 2) 180
= 2(180)
= 360
The sum of the interior angles of a polygon with 5 sides :
= (5 - 2) 180
= 3(180)
= 540
180, 360, 540, .......
It is arithmetic progression with common difference as 180.
19th term will represent sum of interior angles of 21 sided polygon.
an = a + (n - 1) d
a19 = 180 + (19 - 1) 180
a19 = 180 + 18(180)
a19 = 180 + 3240
a19 = 3420
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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