Question

if Sn denotes the sum of first name terms of an AP prove that S12 =3 (s8-s4)

Answer 1

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

S12 = 12/2{ 2a + 11d} = 12a + 66d. ...............1.

S8 = 8/2{2a + 7d} = 8a + 28d ..................2.

S4 = 4/2{2a + 3d} = 4a+ 6d ....................3.

RHS= 3(S8 - S4)
= 3{ 8a + 28d - 4d - 6d}
= 3{ 4a + 22d}
= 12a + 66d
= LHS , as obtained in 1. equation
S12 = 3( S8 - S4 ) is proved.

Answer 2

$\Large \bf Solution :$

Let the first term be a and common difference be d.

We have to prove : $\sf S_{12} = 3(S_{8} - S_{4})$

$\underline {\bf In \: RHS, \: we \: have :}$$\sf 3(S_{8} - S_{4})$

Now,

$\underline{\boxed{\sf S_{n} = \dfrac{n}{2} . [2a+(n-1)d]}}$

$\sf : \implies 3 \Bigg[\dfrac{8}{2}(2a+(8-1)d) - \dfrac{4}{2}(2a+(4-1)d)\Bigg]$

$\sf : \implies 3 \Bigg[\cancel{\dfrac{8}{2}}(2a+7d) - \cancel{\dfrac{4}{2}}(2a+3d)\Bigg]$

$\sf : \implies 3 [4(2a+7d) - 2(2a+3d)]$

$\sf : \implies 3 \times 2[2(2a+7d) - (2a+3d)]$

$\sf : \implies 6(4a+14d - 2a+3d)$

$\sf : \implies 6(2a+11d)$

$\underline {\bf In \: LHS, \: we \: have :}$$\sf S_{12}$

Now,

$\underline{\boxed{\sf S_{n} = \dfrac{n}{2} . [2a+(n-1)d]}}$

$\sf : \implies \dfrac{12}{2} (2a + (12-1)d)$

$\sf : \implies \cancel{\dfrac{12}{2}} (2a + 11d)$

$\sf : \implies 6 (2a + 11d)$

$\bf As, \: LHS = RHS,$

$\underline{\underline{\bf Hence, \: Proved.}}$