FIND THE MISSING TERMS IN ARITHMETIC AND GEOMETRIC SEQUENCE

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

S_n = 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 :

S_n = 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.

(S_n) = 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 (S_n) = a(rⁿ – 1)/r – 1

S₁₀ = 2(2¹⁰ – 1)/2 – 1

S₁₀ = (2^{10 + 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 180^º. 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 :

180^º

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.

a_n = a + (n - 1) d

a₁₉ = 180 + (19 - 1) 180

a₁₉ = 180 + 18(180)

a₁₉ = 180 + 3240

a₁₉ = 3420