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Learn/ 11th Class Mathematics/ Unit 6 /Numericals

Unit 6: Sequences and Series — Exercise 6 3

11th Class Mathematics · Unit 6: Sequences and Series

1.(i) Find A.M. between the given numbers: 2 + √3i, 2 - √3i (ii) Find A.M. between (a+b)² and (a-b)²
Given
(i) Numbers 2 + sqrt{3}i, quad 2 - sqrt{3}i
Formula
Arithmetic Mean A.M. = a+b2
Working
Substitute values A.M. = frac{(2 + sqrt{3}i) + (2 - sqrt{3}i)}{2}
= 42
Result
Arithmetic Mean A.M. = 2
Formula
(ii) Arithmetic Mean A.M. = (a+b)2 + (a-b)22
Expand squares = a2 + 2ab + b2 + a2 - 2ab + b22
Combine like terms = 2(a2 + b2)2
Result
Arithmetic Mean A.M. = a2 + b2
2.If 6, 11, 16 are three A.Ms. between a and b, find a and b.
Given
Three A.Ms. 6, 11, 16 are A.Ms. between a and b
Sequence in A.P. a, 6, 11, 16, b are in A.P.
Find common difference d = 11 - 6 = 5
Find b b - 16 = 5 ⇒ b = 21
Find a 6 - a = 5 ⇒ a = 1
Result
Solution a = 1, quad b = 21
3.Insert five A.Ms. between √2 and 15/√2
Given
First term a1 = sqrt{2}
Last term a7 = frac{15}{sqrt{2}
Formula
Common difference an = a1 + (n-1)d
frac{15}{sqrt{2} - sqrt{2} = 6d ⇒ frac{15 - 2}{sqrt{2} = 6d
6d = frac{13}{sqrt{2} ⇒ d = frac{13}{6sqrt{2}
A₁ A1 = sqrt{2} + d = frac{12}{6sqrt{2} + frac{13}{6sqrt{2} = frac{25}{6sqrt{2}
A₂ A2 = sqrt{2} + 2d = sqrt{2} + frac{13}{3sqrt{2} = frac{19}{3sqrt{2}
A₃ A3 = sqrt{2} + 3d = sqrt{2} + frac{13}{2sqrt{2} = frac{17}{2sqrt{2}
A₄ A4 = sqrt{2} + 4d = sqrt{2} + frac{26}{3sqrt{2} = frac{32}{3sqrt{2}
Result
Five A.Ms. frac{25}{6sqrt{2}, frac{19}{3sqrt{2}, frac{17}{2sqrt{2}, frac{32}{3sqrt{2}, frac{77}{6sqrt{2}
4.The A.M. of two numbers is 7 and their product is 45. Find the numbers.
Given
Arithmetic Mean A.M. = a+b2 = 7 ⇒ a + b = 14 quad (i)
Product ab = 45 quad (ii)
Working
From (ii) a = 45b
Substitute into (i) 45b + b = 14
Simplify 45 + b2 = 14b ⇒ b2 - 14b + 45 = 0
Factor (b - 9)(b - 5) = 0
b = 9 or b = 5
When b = 9 a = 5
When b = 5 a = 9
Result
Solution Numbers are 5 and 9
5.If n arithmetic means are inserted between a and b, prove that d = (b-a)/(n+1), where d is the common difference.
Given
n A.Ms between a and b A1, A2, ldots, An
Sequence in A.P. a, A1, A2, ldots, An, b are in A.P.
Total terms a = a1, quad b = an+2
Formula
an = a1 + (n-1)d
Working
b = a + (n+2-1)d = a + (n+1)d
b - a = (n+1)d
Result
Common difference d = b-an+1
6.If A is the A.M. between a and b, prove that (a-A)² + (A-b)² = 1/2(a-b)²
Given
A is A.M. between a and b a+b2 = A
L.H.S left(a - a+b2right)2 + left(a+b2 - bright)2
= left(2a - a - b2right)2 + left(a + b - 2b2right)2
= left(a-b2right)2 + left(a-b2right)2
= 2left(a-b2right)2 = 12(a-b)2
Result
Therefore L.H.S = R.H.S
7.For what value of n, (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) is the A.M. between a and b, where a ≠ b.
Given
Expression equals A.M. frac{an+1 + bn+1{an + bn} = a+b2
2(an+1 + bn+1) = (a+b)(an + bn)
2an+1 + 2bn+1 = an+1 + abn + an b + bn+1
an+1 + bn+1 = abn + an b
an(a - b) = bn(a - b)
Since a ≠ b an = bn ⇒ left(abright)n = 1 = left(abright)0
Equating exponents: n = 0
Result
Answer n = 0