Question

If S1, S2, S3.......... Sm are the sums of the first n terms of m arithmetic progressions, whose first terms are 1, 4, 9,16, ..... m2 and common differences are 1, 2, 3, 4, .......m respectively, then the value of S1 + S2 + S3 +...........+Sm is:

Answer 1

we know,

Sn = n/2{ 2a + ( n -1)d}

S1 = n/2{ 2× 1 + (n-1)1 } =n( n + 1)/2
S2 =n/2{2×4 + ( n-1)2} =2n(n + 3)/2
S3=n/2{2×9 +( n -1)3} =3n(n+5)/2

……………………………

Sm =n/2{2×m² +( n -1)m} =mn(n+2m-1)/2

now, add all of this ,

S1 + S2 + S3 + S4 +.........Sm = n(n+1)/2 +2n(n+3)/2 +3n( n+5)/2 + 4n(n+7)/2 +..........+ m(n+2m-1)/2

here we use concept of Sigma

Sigma Tm = Sigma {mn/2 + m² -m/2 }

1/2nSigma( m) + Sigma( m²) -1/2Sigma(1)

1/2n{n(n+1)/2} +n(n+1)(2n+1)/6 -n/2

n²(n + 1)/2 + n(n + 1)( 2n +1)/6 -n/2