Chapter 12: Q.43 (page 837)

Use the Binomial Theorem to find the numerical value of
(1.001)⁵ correct to five decimal places.

Short Answer

Expert verified

The numerical value of

(1.001)⁵ correct to five decimal places is $1.00501$

Step by step solution

Step1. Given information

Here(1.001)⁵ is given.

We need to find out the numerical value of (1.001)⁵ correct to five decimal places .

Step 2. Description of expanding (1.001)5

we can rewrite the given expression as (1+10^-3)⁵

now we can apply Binomial Theorem.

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . . . + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} {(1 + 10^{- 3})}^{5} = (\begin{matrix} 5 \\ 0 \end{matrix}) {{(10^{- 3})}^{0} (1)}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {{(10^{- 3})}^{1} (1)}^{4} + (\begin{matrix} 5 \\ 2 \end{matrix}) {{(10^{- 3})}^{2} (1)}^{3} + (\begin{matrix} 5 \\ 3 \end{matrix}) {{(10^{- 3})}^{3} (1)}^{2} + (\begin{matrix} 5 \\ 4 \end{matrix}) {{(10^{- 3})}^{4} (1)}^{1} + (\begin{matrix} 5 \\ 5 \end{matrix}) {{(10^{- 3})}^{5} (1)}^{0} = (\begin{matrix} 5 \\ 0 \end{matrix}) + (\begin{matrix} 5 \\ 1 \end{matrix}) (10^{- 3}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(10^{- 3})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(10^{- 3})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(10^{- 3})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(10^{- 3})}^{5}$

Step3. Finding the numerical value of(1.001)5

The expanded form of (1.001)⁵is

{(1 + 10^{- 3})}^{5} = (\begin{matrix} 5 \\ 0 \end{matrix}) + (\begin{matrix} 5 \\ 1 \end{matrix}) (10

^-3)+52(10-3)2+53(10-3)3+54(10-3)4+55(10-3)5numericalvalueof(1+10-3)5=50+51(10^-3)+52(10-3)2+53(10-3)3+54(10-3)4+55(10-3)5=1+5!4!1!×10-3+5!3!2!×10-6+5!2!3!×10-9+5!1!4!×10-12+1×10-15=1+5×4!1×4!×10-3+5×4×3!1×2×3!×10-6+5×4×3!1×2×3!×10-9+5×4!1×4!×10-12+10-15=1+5×10-3+10×10-6+10×10-9+5×10-12+10-15(1+10-3)5=1+0.005+0.00001+10-8+5×10-12+10-15Numericalvaluescorrecttothe5decimalplacesis1.00501..

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Use the Binomial Theorem to find the numerical value of
(1.001)⁵ correct to five decimal places.

Short Answer

Step by step solution

Step1. Given information

Step 2. Description of expanding (1.001)5

Step3. Finding the numerical value of(1.001)5

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Use the Binomial Theorem to find the numerical value of(1.001)5 correct to five decimal places.

Short Answer

Step by step solution

Step1. Given information

Step 2. Description of expanding (1.001)5

Step3. Finding the numerical value of(1.001)5

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Discrete Mathematics

Geometry

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the Binomial Theorem to find the numerical value of
(1.001)⁵ correct to five decimal places.