evaluate 2/(�ã1+�ã2) + 2/(�ã2+�ã3) + 2/(�ã3+�ã4)... + 2/(�ã8+�ã9)
To evaluate the given expression, we need to simplify each term and then find their sum. Let's break it down step by step:
Step 1: Simplify each term.
The given expression includes terms of the form 2/(�ãn+�ã(n+1)). To simplify these terms, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
For the first term: 2/(�ã1+�ã2)
- Multiply both numerator and denominator by the conjugate of the denominator, which is �ã1-�ã2.
- The term becomes: 2/(�ã1+�ã2) * (�ã1-�ã2)/(�ã1-�ã2)
- Simplifying the numerator, we have 2(�ã1-�ã2).
- The denominator remains the same.
- So the first term simplifies to: 2(�ã1-�ã2) / ((�ã1)^2 - (�ã2)^2)
- Simplifying further, we get: 2(�ã1-�ã2) / (1 - 2)
- The first term becomes: 2(�ã1-�ã2) / (-1)
Similarly, for the second term: 2/(�ã2+�ã3)
- Multiply both numerator and denominator by the conjugate of the denominator, which is �ã2-�ã3.
- The term becomes: 2/(�ã2+�ã3) * (�ã2-�ã3)/(�ã2-�ã3)
- Simplifying the numerator, we have 2(�ã2-�ã3).
- The denominator remains the same.
- So the second term simplifies to: 2(�ã2-�ã3) / ((�ã2)^2 - (�ã3)^2)
- Simplifying further, we get: 2(�ã2-�ã3) / (2 - 3)
- The second term becomes: 2(�ã2-�ã3) / (-1)
We repeat this process for each term in the expression, until we simplify the last term.
Step 2: Simplify the terms and find the sum.
Now, let's evaluate all the terms and find their sum:
Term 1: 2(�ã1-�ã2) / (-1)
Term 2: 2(�ã2-�ã3) / (-1)
Term 3: 2(�ã3-�ã4) / (-1)
.
.
.
Term 7: 2(�ã7-�ã8) / (-1)
Term 8: 2(�ã8-�ã9) / (-1)
Summing up all the simplified terms:
(2(�ã1-�ã2) + 2(�ã2-�ã3) + 2(�ã3-�ã4) + ... + 2(�ã7-�ã8) + 2(�ã8-�ã9)) / (-1)
Simplifying further:
(2�ã1 - 2�ã2 + 2�ã2 - 2�ã3 + 2�ã3 - 2�ã4 + ... + 2�ã7 - 2�ã8 + 2�ã8 - 2�ã9) / (-1)
By canceling out the terms with the same variables:
2�ã1 - 2�ã9
Hence, the simplified expression is 2�ã1 - 2�ã9.