Question

Find the given sum:
1.3 + 5.7 + 9.11 + ...upto n terms.

Answer 1

Above is your answer

Answer 2

Answer:

(4/3) n(2n−1)(2n+1) - n

or

(n/3) ( 16n² - 7)

Step-by-step explanation:

1.3 + 5.7 + 9.11 + ...upto n terms

n term would be (4n - 3)(4n-1)

= 1.3 + 5.7 + 9.11 + .................+.(4n - 3)(4n-1)

= (2-1)(2 + 1) + (6-1)(6+1)  + (10-1)(10+1) +....................................+(4n-2 - 1)(4n-2 + 1)

=( 2² - 1 ) + (6² - 1)  + (10² - 1)  +.........................................+((4n-2)² - 1)

= 2² + 6² + 10² +....................................+ (4n-2)²   - n

= 2²(1² + 3² + 5² + 7² +....................+ (2n-1)²) - n

1² + 3² + 5² + 7² +....................+ (2n-1)² = (1/3)n(2n−1)(2n+1)

= (4/3) n(2n−1)(2n+1) - n

1.3 + 5.7 + 9.11 + ...upto n terms = (4/3) n(2n−1)(2n+1) - n

or

∑ (4n - 3)(4n-1)

= ∑ (16n² - 16n + 3)

= 16 n (n+1)(2n+1)/6 - 16n(n+1)/2 + 3n

= 8 n (n+1)(2n+1)/3 - 8n(n+1) + 3n

=   8 n (n+1)(2n+1)/3  - 8n² - 5n

=  (n/3) ( 8 (n+1)(2n+1)  - 24n - 15)

= (n/3) ( 8(2n² + 3n + 1) - 24n - 15)

= (n/3) ( 16n² + 24n + 8  - 24n - 15)

= (n/3) ( 16n² - 7)