Lecture 27 1. Find 1+2+.. + n by summing the identity (m + 1)² - m² = 2m +1 from m = 1 to n. Similarly find 1² + 2²

+ ... +n² using the identity (m+1)^{a}-m^{2}=3 m^{2}+3 m+1 together with the previous result. 2. John Machin (1680-1751) correctly computed used the identity T to 100 decimal places in 1760. He used the identity \frac{\pi}{4}=4 \tan ^{-1}\left(\frac{1}{5}\right)-\tan ^{-1}\left(\frac{1}{289}\right) Derive Machin's identity. Hint: By using \tan (x+y)=\frac{\tan x+\sin x}{\ln \tan 2 \tan y} and by setting \alpha=\tan ^{-1}\mleft(\frac{1}{5}\mright) you can calculate \tan (2\alpha)=\frac{5}{12},\tan (4\alpha)=\frac{120}{119},\operatorname{and}\tan \mleft(4\alpha-\frac{\pi}{4}\mright)=\frac{1}{239}.