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julsineya [31]
2 years ago
14

Round your answer to the nearest hundredth.

Mathematics
1 answer:
Karolina [17]2 years ago
3 0

Answer:

tam 20 = 4 / x

x = 4 /tan 20 = 4 / .364 = 10.99

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Given that 'n' is a natural number. Prove that the equation below is true using mathematical induction.
LenaWriter [7]

<h3>To ProvE :- </h3>

  • 1 + 3 + 5 + ..... + (2n - 1) = n²

<u>Method</u><u> </u><u>:</u><u>-</u>

If P(n) is a statement such that ,

  1. P(n) is true for n = 1
  2. P(n) is true for n = k + 1 , when it's true for n = k ( k is a natural number ) , then the statement is true for all natural numbers .

\sf\to \textsf{ Let P(n) :  1 + 3 + 5 + $\dots$ +(2n-1) = n$^{\sf 2}$ }

Step 1 : <u>Put </u><u>n </u><u>=</u><u> </u><u>1</u><u> </u><u>:</u><u>-</u><u> </u>

\sf\longrightarrow LHS = \boxed{\sf 1 } \\

\sf\longrightarrow RHS = n^2 = 1^2 = \boxed{\sf 1 }

Step 2 : <u>Assume </u><u>that </u><u>P(</u><u>n)</u><u> </u><u>is </u><u>true </u><u>for </u><u>n </u><u>=</u><u> </u><u>k </u><u>:</u><u>-</u>

\sf\longrightarrow 1 + 3 + 5 + \dots + (2k - 1 ) = k^2

  • Add (2k +1) to both sides .

\sf\longrightarrow 1 + 3+5+\dots+(2k-1)+(2k+1)=k^2+(2k+1)

  • RHS is in the form of ( a + b)² = a²+b²+2ab .

\sf\longrightarrow 1 + 3+5+\dots+(2k-1)+(2k+1)= (k +1)^2

  • Adding and subtracting 1 to LHS .

\sf\longrightarrow 1 + 3+5+\dots+(2k-1)+(2k+1) + 1 -1  = (k +1)^2 \\

\sf\longrightarrow 1 + 3+5+\dots+(2k-1)+(2k+2) - 1 = (k +1)^2

  • Take out 2 as common .

\sf\longrightarrow 1 + 3+5+\dots+(2k-1)+\{2(k+1)-1\}= (k +1)^2

  • P(n) is true for n = k + 1 .

Hence by the principal of Mathematical Induction we can say that P(n) is true for all natural numbers 'n' .

<em>*</em><em>*</em><em>Edits</em><em> are</em><em> welcomed</em><em>*</em><em>*</em>

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