Which of the following form an AP? Justify your answer.
$ \sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots $
Given:
Given sequence is \( \sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots \)
To do:
We have to check whether the given sequence is an AP.
Solution:
In the given sequence,
$a_1=\sqrt3, a_2=\sqrt{12}=\sqrt{4\times3}=2\sqrt3, a_3=\sqrt{27}=\sqrt{9\times3}=3\sqrt{3}, a_4=\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}$
$a_2-a_1=2\sqrt3-\sqrt3=\sqrt3$
$a_3-a_2=3\sqrt3-2\sqrt3=\sqrt3$
$a_4-a_3=4\sqrt3-3\sqrt3=\sqrt3$
Here,
$a_2 - a_1 = a_3 - a_2=a_4 - a_3$
Therefore, the given sequence is an AP.
Related Articles
- Simplify the following: $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$.
- Solve: $3 \sqrt{3}+2 \sqrt{27}+\frac{7}{\sqrt{3}}$.
- Simplify:\( \frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}}+\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}} \)
- Verify that each of the following is an \( \mathrm{AP} \), and then write its next three terms.\( \sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, \ldots \)
- Show that:(i) \( \sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{27 \times 64} \)(ii) \( \sqrt[3]{64 \times 729}=\sqrt[3]{64} \times \sqrt[3]{729} \)(iii) \( \sqrt[3]{-125 \times 216}=\sqrt[3]{-125} \times \sqrt[3]{216} \)(iv) \( \sqrt[3]{-125 \times-1000}=\sqrt[3]{-125} \times \sqrt[3]{-1000} \)
- Fill in the Blanks:(i) \( \sqrt[3]{125 \times 27}=3 \times \)(ii) \( \sqrt[3]{8 \times \ldots}=8 \)(iii) \( \sqrt[3]{1728}=4 \times \)__(iv) \( \sqrt[3]{480}=\sqrt[3]{3} \times 2 \times \sqrt[3]{-} \)(v) \( \sqrt[3]{\square}=\sqrt[3]{7} \times \sqrt[3]{8} \)(vi) \( \sqrt[3]{-}=\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6} \)(vii) \( \sqrt[3]{\frac{27}{125}}=\frac{}{5} \)(viii) \( \sqrt[3]{\frac{729}{1331}}=\frac{9}{-} \)(ix) \( \sqrt[3]{\frac{512}{-}}=\frac{8}{13} \)
- Rationalise the denominator and simplify:\( \frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}} \)
- Simplify each of the following expressions:(i) \( (3+\sqrt{3})(2+\sqrt{2}) \)(ii) \( (3+\sqrt{3})(3-\sqrt{3}) \)(iii) \( (\sqrt{5}+\sqrt{2})^{2} \)(iv) \( (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) \)
- Evaluate the following:$\sqrt[3]{\sqrt{0.000729}}+\sqrt[3]{0.008}$
- Simplify: \( \frac{4 \sqrt{3}}{2-\sqrt{2}}-\frac{30}{4 \sqrt{3}-\sqrt{18}}-\frac{\sqrt{18}}{3-\sqrt{12}} \)
- Which of the following is the compound surd: (a) $4 \sqrt{3}$ (b) $\sqrt{3}$(c) $2 \sqrt[4]{5}$(d) $\sqrt{3}+\sqrt{5}-\sqrt{7}$
- Simplify:\( \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} \)
- Which of the following form an AP? Justify your answer.\( 0,2,0,2, \ldots \)
- Which of the following form an AP? Justify your answer.\( 1,1,2,2,3,3, \ldots \)
- Which of the following form an AP? Justify your answer.\( 11,22,33, \ldots \)
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies