Question

Find Sn if Sn=1.3+2.4+3.5+..............upto n terms​

Answer 1

S=1.3.22+2.4.32+3.5.42+........ upto n terms

Find S?

This was asked under the heading using method of difference and ans given was (n)(n+1)(n+2)(n+3)(2n+3)/10

Answer 2

Answer:

Sₙ = n(n + 1)(2n + 7) / 6

Step-by-step explanation:

Sₙ = 1.3 + 2.4 + 3.5 + .... + (n-1).(n+1) + n.(n+2)

   = 0.2 + 1.3 + 2.4 + 3.5 + .... + (n-1).(n+1) + n.(n+2)

   = (1-1)(1+1) + (2-1)(2+1) + (3-1)(3+1) + (4-1)(4+1) + ... + (n-1)(n+1) + n(n+2)

   = 1²-1² + 2²-1² + 3²-1² + 4²-1² + ... + n²-1²  + n(n+2)

   = ( 1² + 2² + 3² + 4² + ... + n² ) - ( 1 + 1 + 1 + 1 + ... + 1 )  + n(n+2)

   = ( 1² + 2² + 3² + 4² + ... + n² ) - n + n(n+2)

   = n(n + 1)(2n + 1)/6  - n + n² + 2n

   = n(n + 1)(2n + 1)/6  + n² + n

   = n(n + 1)(2n + 1)/6  + n ( n + 1 )

   = n(n + 1) [ (2n + 1)/6 + 1 ]

   = n(n + 1)(2n + 7) / 6

Check:

Try n=4.

S₄ = 1.3 + 2.4 + 3.5 + 4.6 = 3 + 8 + 15 + 24 = 50

The formula above gives:

4(4+1)(2×4+7)/6 = 4(5)(15)/6 = 2(5)(5) = 50

Great, it worked!