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Find the number of terms in the following G.P. (i) 4, 8, 16,…,8192 ?

Find the number of terms in the following G.P.

(i) 4, 8, 16,…,8192 ?

Solution :

Let nth term be "8192"

tn = 8192

a = 4, r = 8/4 = 2

ar n -1 = 8192

4(2)n -1 = 8192

22(2)n -1 = 8192

2 n+1 = 213

n + 1 = 13

n = 12

Hence the 12th term of the above geometric sequence is 8192.

Hence there are 12 terms in the given G.P.

Find the nth term of the following sequences (iii) 3, 8, 13, 18,...

Find the nth term of the following sequences

(iii) 3, 8, 13, 18,...

Solution :

= 3, 8, 13, 18,...

we may see the common things followed in each term.

= (5 - 2), (10 - 2), (15 - 2), (20 - 2),...

nth term of the sequence is 5n - 2.

For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that (i) A U (B ∩ C) = (A U B) ∩ (A U C) (ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(11) For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(iii) Verify using Venn diagrams

Solution:

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S

A U (B ∩ C)

(B ∩ C) = {-1,-2,3,4,5,6} ∩ {-1,2,3,4,5,7}

= {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}

= {-3,-1,0,3,4,5,6,8,10} --- (1)

R.H.S

(A U B) ∩ (A U C)

(A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}

= {-3,-2,-1,0,3,4,5,6,8,10}

(A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}

= {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}

= {-3,-1,0,3,4,5,6,8,10} --- (2)

(1) = (2)

A U (B ∩ C) = (A U B) ∩ (A U C)

Hence proved

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

L.H.S

(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}

= {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}

= {-1,4,6} ---- (1)

R.H.S

(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}

= {-1,4,6}

(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}

= {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}

= {-1,4,6} ---- (2)

(1) = (2)

A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Hence proved

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find (i) A U (B ∩ C) (ii) A ∩ (B U C) (iii) A \ (C \ B)

Question 4 :

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find

(i) A U (B ∩ C) (ii) A ∩ (B U C) (iii) A \ (C \ B)

Solution :

(i) A U (B ∩ C)

(B ∩ C) = {2, 4, 6} ∩ {1, 2, 3, 4, 5, 6}

= {2, 4, 6}

A U (B ∩ C) = {4, 6, 7, 8, 9} U {2, 4, 6}

= {2, 4, 6, 7, 8, 9}

(ii) A ∩ (B U C)

(B U C) = {2 ,4, 6} U {1, 2, 3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

A ∩ (B U C) = {4, 6, 7, 8, 9} ∩ {1, 2, 3, 4, 5, 6}

= {4, 6}

(iii) A \ (C \ B)

C \ B = {1, 2, 3, 4, 5, 6} \ {2, 4, 6}

= {1, 3, 5}

A \ (C \ B) = {4, 6, 7, 8, 9} \ {1, 3, 5}

= {4, 6, 7, 8, 9}