Question

The Sum of n terms of an A.P. is 5n^2 – 3n. Find the A.P. Hence find its 10th term.

Answer 1

EXPLANATION.

Sum of n terms of an A.P = 5n² - 3n.

As we know that,

⇒ Tₙ = Sₙ - Sₙ₋₁.

Using this formula in equation,we get.

⇒ 5n² - 3n - [5(n - 1)² - 3(n - 1)].

⇒ 5n² - 3n - [5(n² + 1 - 2n) - 3n + 3].

⇒ 5n² - 3n - [5n² + 5 - 10n - 3n + 3].

⇒ 5n² - 3n - 5n² - 5 + 10n + 3n - 3.

⇒ - 5 + 10n - 3.

⇒ 10n - 8.

⇒ 10n - 8 = Algebraic expression.

Put n = 1 in equation, we get.

⇒ 10(1) - 8.

⇒ 10 - 8.

⇒ 2.

Put n = 2 in equation, we get.

⇒ 10(2) - 8.

⇒ 20 - 8.

⇒ 12.

Put n = 3 in equation, we get.

⇒ 10(3) - 8.

⇒ 30 - 8.

⇒ 22.

Put n = 4 in equation, we get.

⇒ 10(4) - 8.

⇒ 40 - 8.

⇒ 32.

Series = 2, 12, 22, 32.‎.‎.‎.‎.‎.‎.‎.‎‎.‎

First term = a = 2.

Common difference = d = b - a = c - b.

Common difference = d = 12 - 2 = 22 - 12.

Common difference = d = 10.

As we know that,

General term of an A.P.

⇒ Tₙ = a + (n - 1)d.

⇒ T₁₀ = a + (10 - 1)d.

⇒ T₁₀ = a + 9d.

⇒ T₁₀ = 2 + 9(10).

⇒ T₁₀ = 2 + 90.

⇒ T₁₀ = 92.

                                                                                                                           

MORE INFORMATION.

Supposition of terms in A.P.

(1) = Three terms as : a - d, a, a + d.

(2) = Four terms as : a - 3d, a - d, a + d, a + 3d.

(3) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.

Answer 2

$\Large{\underbrace{\underline{\sf{Understanding\; the\; Question}}}}$

Here in this question, concept of Arithmetic Progression is used. We are given sum of n terms of an AP and we have to find the AP and it's 10th term. It is also advantageous that we are given sum of n terms in the form of only n and numerical values. So we can find the AP by following steps:

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Step 1:

~Find first term

We have given sum of n terms of AP:

$\underline{\boxed{\sf{S_n=5n^2-3n}}}\bigstar$

Now put value of n to be 1.

$\sf\hookrightarrow S_1=5(1)^2-3(1)$

Now simply it!!

$\sf\hookrightarrow S_1=5-3$

$\sf\hookrightarrow S_1=2$

Here we have obtained sum of 1 term of AP. It is obvious that sum of 1 terms of AP will comprise only 1st term.

So first term of AP (A1) is 2.

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Step 2:

~Find 2nd term.

Put value of n=2 in the given expression.

$\sf \hookrightarrow S_n=5n^2-3n$

$\sf\hookrightarrow S_2=5(2)^2-3(2)$

Now simply it!!

$\sf\hookrightarrow S_2=5(4)-3(2)$

$\sf\hookrightarrow S_2=20-6$

$\sf\hookrightarrow S_2=14$

So, sum of first 2 terms of AP is 14.

Sum of first 2 terms will comprise 1st term and 2nd term.

i.e:-

$\sf: \implies S_2=A_1+A_2$

Now put value of S2 and A1 to find A2!

$\sf\hookrightarrow 14=2+A_2$

$\sf\hookrightarrow 14-2=A_2$

$\sf\hookrightarrow 12=A_2$

So 2nd term of AP (A2) is 12.

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Step 3:

~Find common difference

Now we have 1st and 2nd term, so we can find common difference of AP.

⇒ Common difference=A2-A1

⇒ Common difference=12-2

⇒ Common difference=10

Please note that common difference can be find by subtracting either of 2 consecutive terms of AP.

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Step 4:

~ Now form AP

Now we have obtained the common difference and first term of AP, so we can find the AP by adding common difference in particular term to get the next term.

So let's find 2 more terms to justify AP.

⇒ 2nd term=12

⇒ 3rd term=2nd term+common difference

⇒ 3rd term=12+10

⇒ 3rd term=22

Now 4th term can be find by adding common difference in 3rd term.

⇒ 4th term=3rd term+common difference

⇒ 4th term=22+10

⇒ 4th term=32

So the required AP is:

2,12,22,32...

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Now we are asked to find 10th term of AP.

Nth term can be defined as:

An=A1+(n-1)d

By putting value of n=10, we can find 10th term.

⇒ A10=2+(10-1)10

⇒ A11=2+(9)10

⇒ A10=2+90

⇒ A10=92

So the required 10th term of AP is 92.

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