What is \(\frac{1}{\sqrt{10}+\sqrt{9}} + \frac{1}{\sqrt{11}+\sqrt{10}} + \frac{1}{\sqrt{12}+\sqrt{11}} + \dots + \frac{1}{\sqrt{196}+\sqrt{195}}\) equal to ?
Understanding the Series Sum Problem
The question asks us to find the sum of a specific series involving square roots. The series looks like this:
\( S = \frac{1}{\sqrt{10}+\sqrt{9}} + \frac{1}{\sqrt{11}+\sqrt{10}} + \frac{1}{\sqrt{12}+\sqrt{11}} + \dots + \frac{1}{\sqrt{196}+\sqrt{195}} \)
This type of series often simplifies using a technique called rationalization, leading to a telescoping sum.
Step 1: Rationalizing the General Term
Let's look at a general term in the series, which is of the form \(\frac{1}{\sqrt{n+1}+\sqrt{n}}\). To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{n+1}-\sqrt{n}\).
\( \frac{1}{\sqrt{n+1}+\sqrt{n}} = \frac{1}{\sqrt{n+1}+\sqrt{n}} \times \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+1}-\sqrt{n}} \)
Using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) in the denominator:
\( = \frac{\sqrt{n+1}-\sqrt{n}}{(\sqrt{n+1})^2 - (\sqrt{n})^2} \)
\( = \frac{\sqrt{n+1}-\sqrt{n}}{(n+1) - n} \)
\( = \frac{\sqrt{n+1}-\sqrt{n}}{1} \)
\( = \sqrt{n+1}-\sqrt{n} \)
So, each term \(\frac{1}{\sqrt{n+1}+\sqrt{n}}\) in the series is equivalent to \(\sqrt{n+1}-\sqrt{n}\).
Step 2: Applying Rationalization to the Series
Now we can rewrite each term in the given series using the simplified form:
- The first term corresponds to \(n=9\): \(\frac{1}{\sqrt{10}+\sqrt{9}} = \sqrt{10}-\sqrt{9}\)
- The second term corresponds to \(n=10\): \(\frac{1}{\sqrt{11}+\sqrt{10}} = \sqrt{11}-\sqrt{10}\)
- The third term corresponds to \(n=11\): \(\frac{1}{\sqrt{12}+\sqrt{11}} = \sqrt{12}-\sqrt{11}\)
- ...
- The last term corresponds to \(n=195\): \(\frac{1}{\sqrt{196}+\sqrt{195}} = \sqrt{196}-\sqrt{195}\)
Step 3: Identifying the Telescoping Sum
Let's write out the series with the simplified terms:
\( S = (\sqrt{10}-\sqrt{9}) + (\sqrt{11}-\sqrt{10}) + (\sqrt{12}-\sqrt{11}) + \dots + (\sqrt{196}-\sqrt{195}) \)
Notice that the positive part of each term cancels out the negative part of the next term. This is called a telescoping sum.
Let's expand it slightly to see the cancellation:
\( S = -\sqrt{9} + \sqrt{10} \)
\( \quad - \sqrt{10} + \sqrt{11} \)
\( \quad - \sqrt{11} + \sqrt{12} \)
\( \quad \dots \)
\( \quad - \sqrt{195} + \sqrt{196} \)
After cancellation, we are left with only the first part of the first term (negative) and the second part of the last term (positive).
\( S = -\sqrt{9} + \sqrt{196} \)
Step 4: Calculating the Final Value
Now, we just need to calculate the square roots and find the difference:
- \(\sqrt{9} = 3\)
- \(\sqrt{196} = 14\)
So, the sum S is:
\( S = 14 - 3 \)
\( S = 11 \)
Therefore, the value of the given series is 11.
Similar Questions
- What is \(\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + ...\) equal to ?
- There are \(n\) concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements :
I. The diagonal of the \(n\)th square is \(2n+\sqrt{2}-2\)
II. The area included between \(n\)th square and \((n-1)\)th square is independent of \(n\)
Which of the statements given above is/are correct?
Important Questions from Sequence & Series
Which of the following options should come in place of the question mark (?) in the given series based on the English alphabetical order?
SHK, WLO, APS, ETW,?
Which of the following options should come in place of the question mark (?) in the given series based on the English alphabetical order?
EGI, KMO, QSU, WYA,?
Which of the following options should come in place of the question mark (?) in the given series?
14 15 18 23 30?
Two sets of numbers are given below. In each set of numbers, certain mathematical operation(s) on the first number result(s) in the second number. Similarly, certain mathematical operation(s) on the second number result(s) in the third number and so on. Which of the given options follows the same set of operations as in the given sets? (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into their constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
What should come in place of the question mark (?) in the given series based on the English alphabetical order?
TAW VCY XEA ZGC ?
