If an arithmetic sequence {aₙ} has all terms positive and the first term and the common difference equal, it satisfies-
[tex]\text{$\rm\displaystyle \sum^{15}_{k=1}\dfrac{1}{\sqrt{{a}_{k}}+\sqrt{{a}_{k+1}}}=2$}[/tex]
-then what is the value of a₄?
[tex]\;[/tex]
Given,
[tex]\begin{cases}&\rm {a}_{1}=a=d\\&\rm {a}_{n} > 0\\&\rm d > 0\end{cases}[/tex]
Consider,
[tex]\text{$\rm\dfrac{1}{\sqrt{{a}_{k}}+\sqrt{{a}_{k+1}}}$}[/tex]
[tex]\boxed{\textrm{$\bigstar$ Rationalize the denominator.}}[/tex]
[tex]\text{$\rm =\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{({a}_{k})-({a}_{k+1})}$}[/tex]
An arithmetic sequence is equal in difference.
[tex]\boxed{\rm \Longrightarrow {a}_{k+1}-{a}_{n}=d}[/tex]
Now consider,
[tex]\text{$\rm =\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{-d}$}[/tex]
By summation rule,
[tex]\text{$\rm\displaystyle \sum^{15}_{k=1}\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{-d}$}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{1}{d}\sum^{15}_{k=1}(\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}})$}[/tex]
[tex]\boxed{\textrm{$\bigstar$ Telescoping series.}}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{\sqrt{d}-\sqrt{16d}}{d}$}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{(1-4)\sqrt{d}}{d}$}[/tex]
[tex]\text{$\rm\displaystyle =\dfrac{3\sqrt{d}}{d}$}[/tex]
[tex]\text{$\rm\displaystyle 2=\dfrac{3\sqrt{d}}{d}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle 2=\dfrac{3}{\sqrt{d}}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle \sqrt{d}=\dfrac{3}{2}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle {(\sqrt{d})}^{2}={\bigg(\dfrac{3}{2}\bigg)}^{2}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle d=\dfrac{9}{4}$}[/tex]
Now consider-
[tex]\text{$\rm {a}_{4}$}[/tex]
[tex]\text{$\rm =a+(4-1)d$}[/tex]
[tex]\text{$\rm =4d$}[/tex]
[tex]\text{$\rm =9$}[/tex]
(Thanks for loving Korea! Here are some mathematical terms for you.)
[Words from the question.]
rationalization- 유리화
common- 공통의
arithmetic- 산술
sequence/progression- 수열
series- 수열의 합
term- 항
difference- 차
sum- 합
square- 제곱
root- 근
telescoping series- 망원급수
[Words of the topic.]
geometric sequence- 기하수열/등비수열
harmonic sequence- 조화수열
ratio- 비
Have a good day, and keep learning!
Answer:
Oh..!
Such a cute words in 'Handi'.
Never mind...
It's a common Thing... in Every country..
I am fine..!
Why you are not..?
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Answers & Comments
Verified answer
If an arithmetic sequence {aₙ} has all terms positive and the first term and the common difference equal, it satisfies-
[tex]\text{$\rm\displaystyle \sum^{15}_{k=1}\dfrac{1}{\sqrt{{a}_{k}}+\sqrt{{a}_{k+1}}}=2$}[/tex]
-then what is the value of a₄?
[tex]\;[/tex]
Solution
Given,
[tex]\begin{cases}&\rm {a}_{1}=a=d\\&\rm {a}_{n} > 0\\&\rm d > 0\end{cases}[/tex]
[tex]\;[/tex]
Consider,
[tex]\text{$\rm\dfrac{1}{\sqrt{{a}_{k}}+\sqrt{{a}_{k+1}}}$}[/tex]
[tex]\boxed{\textrm{$\bigstar$ Rationalize the denominator.}}[/tex]
[tex]\text{$\rm =\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{({a}_{k})-({a}_{k+1})}$}[/tex]
[tex]\;[/tex]
An arithmetic sequence is equal in difference.
[tex]\boxed{\rm \Longrightarrow {a}_{k+1}-{a}_{n}=d}[/tex]
[tex]\;[/tex]
Now consider,
[tex]\text{$\rm =\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{-d}$}[/tex]
[tex]\;[/tex]
By summation rule,
[tex]\text{$\rm\displaystyle \sum^{15}_{k=1}\dfrac{\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}}}{-d}$}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{1}{d}\sum^{15}_{k=1}(\sqrt{{a}_{k}}-\sqrt{{a}_{k+1}})$}[/tex]
[tex]\boxed{\textrm{$\bigstar$ Telescoping series.}}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{\sqrt{d}-\sqrt{16d}}{d}$}[/tex]
[tex]\text{$\rm\displaystyle =-\dfrac{(1-4)\sqrt{d}}{d}$}[/tex]
[tex]\text{$\rm\displaystyle =\dfrac{3\sqrt{d}}{d}$}[/tex]
[tex]\;[/tex]
Given,
[tex]\text{$\rm\displaystyle 2=\dfrac{3\sqrt{d}}{d}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle 2=\dfrac{3}{\sqrt{d}}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle \sqrt{d}=\dfrac{3}{2}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle {(\sqrt{d})}^{2}={\bigg(\dfrac{3}{2}\bigg)}^{2}$}[/tex]
[tex]\text{$\rm\Longrightarrow\displaystyle d=\dfrac{9}{4}$}[/tex]
[tex]\;[/tex]
Now consider-
[tex]\text{$\rm {a}_{4}$}[/tex]
[tex]\text{$\rm =a+(4-1)d$}[/tex]
[tex]\text{$\rm =4d$}[/tex]
[tex]\text{$\rm =9$}[/tex]
[tex]\;[/tex]
Learn More
(Thanks for loving Korea! Here are some mathematical terms for you.)
[Words from the question.]
rationalization- 유리화
common- 공통의
arithmetic- 산술
sequence/progression- 수열
series- 수열의 합
term- 항
difference- 차
sum- 합
square- 제곱
root- 근
telescoping series- 망원급수
[Words of the topic.]
geometric sequence- 기하수열/등비수열
harmonic sequence- 조화수열
ratio- 비
[tex]\;[/tex]
Have a good day, and keep learning!
Answer:
Oh..!
Such a cute words in 'Handi'.
Never mind...
It's a common Thing... in Every country..
I am fine..!
Why you are not..?