Question

Prove that 1/2 - 1/3 + 1/4 - 1/5 + ....... + 1/98 - 1/99 + 1/100 > 1/5.

Answer 1

Let S1=12+14+16+...... +1100S2=13+15+17+....... +199So we have to find S1- S22S1=22+24+26+...... +2100⇒2S1=1+12+13+14+...... +150⇒2S1-1=12+13+14+...... +150AndS1+S2=12+13+14+15+16+...... +199+1100⇒S1+S2=2S1-1+151+152+.... +199+1100⇒2S1-S1- S2=1-151+152+....+199+1100S1- S2=1-151+152+.... +199+1100 ; equation (i) .Now 199+1100<16500197+198<16500....Hence equation (i) becomes⇒S1- S2>1-16500+16500+16500+....25 times⇒S1- S2>1-16×25500⇒S1-S2>1-45⇒S1- S2>15⇒12-13+14-15+16+......-199+1100>15 Hence proved