Question Number 104821 by bobhans last updated on 24/Jul/20

3+8+15+24+35+...   find sum of 50^(th) −term

Answered by bemath last updated on 24/Jul/20

S_(50)  = 3C_1 ^( 50)  + 5C _2^(50)  + 2C _3^(50)           = 150 +5×25×49+2×((50×49×48)/(3×2×1))          = 45475 ★

Commented bybobhans last updated on 24/Jul/20

thank you

Answered by 1549442205PVT last updated on 24/Jul/20

Since 8−3=5,15−8=7,24−15=9,35−24=11  and 5,7,9,11...form an arithmetic progress,  u_n =an^2 +bn+c,so   { ((u_1 =3=a+b+c(1))),((u_2 =8=4a+2b+c(2)⇒ { ((3a+b=5(4))),((5a+b=7(5))) :})),((u_3 =15=9a+3b+c(3))) :}  ⇒2a=2⇒a=1⇒b=2,c=0  u_n =n^2 +2n⇒u_(50) =50^2 +2×50=2600  S_n =Σ_(k=1) ^(n) (k^2 +2k)=Σ_(k=1) ^(n) k^2 +2Σ_(l=1) ^(n) k=((n(n+1)(2n+1))/6)+n(n+1)  =((n(n+1)(2n+7))/6).Therefore,  S_(50) =((50×51×107)/6)=25×17×107=45475

Commented bybobhans last updated on 24/Jul/20

sir the question want to find Σ_(n=1) ^(50) u_n   o yes..thank you