Dear srishti

sum of nterms can be written as  as n+n-1/2 + n-2/4-----+1/2n-1

which is sum of nterms of an AGP

The sum Sn of first n terms of an A.G. P. is obtained in the   following way :

 

Sn = ab + (a + d)br + (a + 2d)br2 +.........+(a + (n - 2)d)brn-2 + (a + (n - 1)d)brn-1
 

Multiply both sides by r, so that
 

r Sn = abr + (a + d)br2+.........+(a + (n - 3)d)brn-2 + (a + (n - 2)d)brn-1 + (a + (n - 1)d)brn
 

Subtracting, we get
 

(1 - r)Sn = ab + dbr + dbr2 +.......+dbrn - 2 + dbrn - 1 - (a + (n - 1)d)brn
 

= ab + dbr/(1–rn–1)/(1–r) – (a+(n–1)d)brn => Sn = ab/1–r + dbr(1–r^n–1)/(1–r)2 – (a+(n–1)d)brⁿ/1–r
 
a=n

b=1

r=.5

d=-1

So Sn = 2n+(-1)*0.5*1(1-0.5^n-1)*4-2ⁿ

=>Sn = 2n-2(1-0.5^n-1)-2ⁿ

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