(i)If t n = 3 n – 2 n then find 
.
(ii)If t n = n(n + 2) then find 
.
(iii)Find the sum to n terms of the series 1 2 – 2 2 + 3 2 – 4 2 + 5 2 – 6 2 +.....
(iv)10 2 + 13 2 + 16 2 + ...... upto 10 terms
(v) If 
, then find the 
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) 
(3 k+1 + 1) – 2 k+1 (ii) 
k(k + 1) (2k + 7)
(iii) – 
if n is even, 
if n is odd
(iv) 6265 (v) 
Sol. (i) T n = 3 n – 2 n ; S n = 
– 
S n = 
– 
; S n = 
– 2 n+1
(ii) 
= 
= 
k(k + 1) (2k + 7)
(iii) clearly n 6 term of the given series is negative or positive accordingly as n is even or odd respectively
n is even
1 2 – 2 2 + 3 2 – 4 2 + 5 2 – 6 2 + ....... + (n – 1) 2 – n 2
= (1 2 – 2 2 ) + (3 2 – 4 2 ) + (5 2 – 6 2 ) + ..... ((n – 1) 2 – n 2 )
= (1 – 2) (1 – 2) + (3 – 4) (3 + 4) + (5 – 6) (5 + 6) + ...... + ((n –1) – (n)) (n – 1 + n) = – 
n is odd
(1 2 – 2 2 ) + (3 2 – 4 2 ) + ..... {(n – 2) 2 – (n – 1) 2 } + n 2
= (1 – 2) (1 + 2) + (3 – 4) (3 + 4)+ ........+ [(n – 2) – (n – 1)) [ (n–2) + (n – 1)] + n 2
= –(1 + 2 + 3 + 4 +..........+ (n–2) + (n – 1)) + n 2
= 
= 
(iv) 
= 6265
(v) S n = 
= n(2n 2 + 9n + 13)
⇒ I (r) = S r – S r–1
= r(2r 2 + 9r + 13) – (r – 1) (2 (r – 1) 2 + 9(r – 1) + 13)
= 6r 2 + 12r + 6 – 6 (r + 1) 2
⇒ 
= 
⇒ 
= 
= 
= 
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems