Question

If Sn denotes the sum of first n terms of an arithmetic progression.

Then prove that S30 = 3[S20 – S10]​

Answer 1

Step-by-step explanation:

Given:

• Sⁿ denotes the sun of first n term of AP.

To Prove:

• S³⁰ = 3[S²⁰ – S¹⁰]

Solution: We know that the sum of n terms of an AP is given by

★ Sⁿ = n/2 × {2a + (n – 1)d} ★

Let assume that here

• First term = a
• Common difference = d

Using above formula lets find the sum of 30th term of AP.

➟ S³⁰ = 30/2 × {2 × a + (30 – 1)d}

➟ S³⁰ = 15 × {2a + 29d}

➟ S³⁰ = 30a + 435d.......(eqⁿ 1)

Now 20th term of AP

➟ S²⁰ = 20/2 × {2 × a + (20 – 1)d}

➟ S²⁰ = 10 × {2a + 19d}

➟ S²⁰ = 20a + 190d

Now finding 10th term of AP

➟ S¹⁰ = 10/2 × {2 × a + (10 – 1)d}

➟ S¹⁰ = 5 × {2a + 9d}

➟ S¹⁰ = 10a + 45d

A/q

• S³⁰ = 3[S²⁰ – S¹⁰]

$\implies{\rm }$ 3{20a + 190d – (10a + 45d)}

$\implies{\rm }$ 3{20a + 190d – 10a – 45d}

$\implies{\rm }$ 3{10a + 145d}

$\implies{\rm }$ 30a + 435d.......(eqⁿ 2)

Here we got i.e Equation 1 = 2

Hence, S³⁰ = 3[S²⁰ – S¹⁰]

$\large\bold{\texttt {Proved }}$

Answer 2

Answer:

I hope this helps you

Step-by-step explanation:

the formula for the sum of an AP is

S/2(2a+(n-1)d)