Question

1.2+3.4+5.6...+(2n-1)2n=¹/³n(n+1)(4n+1)

Answer 1

Answer:

first take1 3 5 7 as an APTHEN the another onefind term of the whole AP and solve accordingly

Answer 2

Answer:

Step-by-step explanation:

Please check the question

Clearly,general term(nth term) of the sequence = (2n-1)2n

∴ L.H.S. =1.2 +3.4+5.6+...+(2n-1).2n =  $\sum_{r=1}^{n}$$(2r-1)\cdot 2r$

= $\sum_{r=1}^{n}$(4r² - 2r)

= 4. $\frac{n\cdot (n+1)\cdot (2n+1)}{6}$ - 2$\frac{n\cdot (n+1)}{2}$              ( $\sum_{r=1}^{n}$ $n^{2}$ = $\frac{n\cdot (n+1)\cdot (2n+1)}{6}$ )

= 2.$\frac{n\cdot (n+1)\cdot (2n+1)}{3}$ - n(n+1)                  ( $\sum_{r=1}^{n}$ n = $\frac{n\cdot (n+1)}{2}$ )

=$n\cdot (n+1)$ ( 2.$\frac{2n+1}{3}$ -1)

= $n\cdot (n+1)$($\frac{4n+2-3}{3}$)

=$\frac{n\cdot (n+1)\cdot (4n-1)}{3}$

= R.H.S.