Question

the series 2/1²-3/2²+4/3²-5/4²+.....is......​

Answer 1

Step-by-step explanation:

12−22+......................

When n is even

⟹(1+2)(1−2)+(3+4)(3−4)..............

⟹−(3+7+11+......)

It is an Arthematic Progression with first term 3 and common difference 4

⟹−(m2(2.3+(m−1)4)

⟹−(m2(2+4.m))

⟹−(m.2.(2.m+1)2)

⟹−(m.(2.m+1))

Each term in this AP corresponds to 2 terms in the original question hence m=n/2

⟹−(n2(2.n2+1)

⟹−(n(n+1)2)

When n is odd

The sum to first n-1 terms can be achieved by the above formula

we get

⟹−(n−1).n2

adding n2 to this we get

⟹−(n−1).n2+n2

⟹n(n−n−12)

⟹n.(n+1)2

Hence answer be written as (−1)n+1(n.(n+1)2)