AP EAMCET202423 May 2024Morning ShiftMathematicsSequences and SeriesActual
The n^ th term of the series 1+(3+5+7)+(9+11+13+15+17)+
Options
- A(2 n+1) [n^2-(n-1)^2 ]
- B(2 n-1) [(n-1)^2-n^2 ]
- C(2 n+1) [(n-1)^2-n^2 ]
- D(2 n-1) [(n-1)^2+n^2 ]
Correct answer
D. (2 n-1) [(n-1)^2+n^2 ]
Step-by-step solution
Given the series 1 _ t₁ + (3+5+7) _ t₂ + (9+11+13+15+17) _ t₃ + . Number of numbers in n^ th term =2 n-1 Let first term in t_ n =a_ n So, t_n= a, a+2, a+4, . upto (2 n-1) terms Since, a₁=1, a₂=3, a₃=9, a₄=19 as so on, we get a_ n =2(n-1)^2+1 Now, aligned & t_n= a_n+a_n+2+a_n+4+ . . upto (2 n-1) term & = (2 n-1) 2 2 a_n+(2 n-1-1) 2 & =(2 n-1) (2(n-1)^2+1+2 n-2 ) & =(2 n-1) (2 n^2-2 n+1 ) & t_n=(2 n-1) ((n-1)^2+n^2 ) aligned