11+√2+1√2+√3+1√3+√4+…+1√15+√16=?
Answer:
3
- If we multiply the numerator and denominator of the first term by √2−1 we get:
11+√2=(√2−1)(√2+1)(√2−1)=√2−12−1[Since, a2−b2=(a+b)(a−b)]=√2−1 - Similarly, if we multiply numerator and denominator of the second term by √3−√2, we get:
1√2+√3=√3−√2(√3+√2)(√3−√2)=√3−√23−2[Since, a2−b2=(a+b)(a−b)] =√3−√2 - Using the same technique with the other terms too, we can re-write the expression in question as:
11+√2+1√2+√3+1√3+√4+…+1√15+√16=(√2−1)+(√3−√2)+(√4−√3)+…+(√16−√15)=−1+√16=−1+4=3