Find the `n^(th)` term and sum to n terms of the series `8, 14, 22, 32, 44,.....`

Explanation

Observe the successive differences
`{:(8,,14,,22,,32,,44,),(,6,,8,,10,,12,,"1st order difference"Deltat),(,,2,,2,,2,,,"2nd order difference"Delta^2t):}`
As 2nd order differences are constant the nth term , ` t_(n)` is polynomial in n of degree 2.
Let `t_(n) = an^(2) + bn + c `
Now ` t_(1) = a + b + c = 8 ` ...(i)
`t_(2) = 4a + 2b + c = 14 ` ...(ii)
` t_(3) = 9a + 3b + c = 22` ...(iii)
From (i) and (ii) by subtraction ,
` 3a + b = 6 ` (iv)
From (ii) and (iii) by subtraction .
` 5a + b = 8 ` ..(v)
From (iv) and (v) upon solving a = 1 , b = 3 . Then from (i) ,
c = 4
And then ` t_(n) = n^(2) + 3n + 4 `