KACL-DS-INC

--> Produced by MATCAL Program 

--> The sum of the nth term in an A.P is given by the equation 

below 

--> S(n) = n / 2 [ 2a + ( n - 1 ) x d ] 

--> S(n) = Sum of the nth term,  n = number of terms, a = first 

term, d = common difference 

--> S(n) = 327, n = 24

--> 327 = 24 / 2 ( 2a  + ( 24 - 1 ) x d ) 

--> 327 = 12 ( 2a  +  ( 23)  x d ) 

--> 327 = 12 ( 2a  +  ( 23) d ) 

--> 327 = 24a + 276d 

--> 24a + 276d  = 327  eq ( 1 ) 

--> Produced by MATCAL Program 

--> The second equation is given below 

--> S(n) = 383, n = 3

--> 383 = 3 / 2 ( 2a  + ( 3 - 1 ) x d ) 

--> 383 = 1.5 ( 2a  +  ( 2)  x d ) 

--> 383 = 1.5 ( 2a  + 2d ) 

--> 383 = 3a + 3d 

--> 3a + 3d  = 383  eq ( 2 ) 

--> ------------------------------------------------------

--> 24a + 276d  = 327  eq ( 1 ) 

--> 3a + 3d  = 383  eq ( 2 ) 

-- > ------------------------------------------------------

--> Using Simultaneous equation of substitution method 

-- > 24a  + 276d   = 327 eq (1) 

-- > 3a  + 3d   = 383 eq (2) 

-- > 24a  = 327 - 276d  eq (3) 

--> Substitute eq(3) into eq(2) for a 

-- > a  = ( 327 - 276d ) / 24 eq (4) 

--> 3 x ( 327 - 276d ) / 24 + 3d = 383

--> 3 x ( 327 - 276d )  + 72d = 9192

--> 981 - 828d + 72d = 9192

--> -756d = 8211

--> d = 8211 / -756

--> d = -10.8611111111111

--> The common difference (d) = -10.8611111111111

--> From equation 4 above 

-- > a  = ( 327 - 276d ) / 24 eq (4) 

-- > Substitute the value of d into equation 4 to get a

-- > 24a  = ( 327 - 276 x 8211 / -756 )

-- > 24a  = ( 327 - 2266236 / -756 )

-- > 24a  = ( 327 + 2997.66666666667 )

--> a = 3324.66666666667 / 24