Here, l = 28, S = 144
n = 9
l = a + (n – 1)d
28 = a + (9 – 1)d
⇒ 28 = a + 8d ...(i)

And,

      

    32 = 2(a + 4d)
    a + 4d = 16                                  .....(ii)
Solving (i) and (ii) we get,
      a = = 32 -28 = 4
Hence ,    a   =  4

Here,  a_n = 4,     d = 2
S_n = –14
We know that,  

    a_n  = a + (n -1)d
  4 = a + (n -1) x 2
  4 = a + 2n - 2
  a + 2n = 6  a = 6 - 2 n    ....(i)

And.     Sn =   [2a+ (n - 1)d]
           - 14 = 
            - 28 = n[2a + 2 (n -1) ]
            - 28 = 2n [a + (n -1)]
            - 14 = n[a + (n - 1)]      ......(ii)

Putting the value of (i) in (ii), we get,
– 14 = n [6 – 2n + n – 1]
⇒ – 14 = n[5 – n]
⇒ –14 = 5n – n²
⇒ n² –5n – 14 = 0
⇒ (n – 7) (n + 2) = 0
⇒ n – 7 = 0 or n + 2 = 0
⇒ n = 7 or n = –2
But n cannot be –ve. ∴ n = 7.
Putting this value of V in equation (i), we get
a + 2 x 7 = 6
⇒ a = 6 – 14 = – 8
Hence, n = 7 and a = –8

Here,    a = 2, d = 8, S_n = 90
We know that,

            S_n =  [2 x 2 + (n -1)d]
     180  =n[2 x 2 (n - 1) x 8]
     180 = n(8n - 4)
     180 = 8n² - 4n
     8n² - 4n - 180 = 0
     2n² - n - 45 = 0     [Dividing by 4]
    2n(n - 5) (2n + 9) = 0
    n - 5 = 0  or 2n + 9 = 0

    n = 5 or n = 
But number of terms cannot be (- ve)


Now,   a₅ = a+ 4d
              = 2 + 4 + 8
Hence,     n = 5      
a_n = 34

Here a = 3, n = 8
S = 192
We know that,  
      

   6 +7d = 48

Hence, d = 6 Ans.

Here a = 8

S_n = 210
We know that, a_n = a + (n - 1)d
   62 - 8 = (n - 1)d
   54 = (n -d)d                   ...(i)

And,  S_n =  '[ 2a - (n - 1) d
  210 = [ 2 x 8 + (n - 1) d]
  420 = n[16 +(n - 1)d] .......(ii)
From (i) and (ii), we get
420  = n[ 16 + 54]
   420  = 70n

Putting this value of n in (i) we get
54 = (6 -1) x d
     5d = 54

     d = 
Hence,  n = 6 and d =