Question

If the sum of the first n terms of an AP is Sn = 5n - n2 what is the first term? What is the sum
of first two terms ? What is the second term ? Similarly find the 3rd, the 10th and the nth terms.​

Answer 1

Answer:

★ first term = 4

★ Sum of first two terms = 6

★ second term = 2

★ 3rd term = 0

★ 10th term = -14

★ nth term = 6 - 2n

Step-by-step explanation:

Given :

the sum of the first n terms of an AP is Sₙ = 5n - n²

To find :

• the first term
• the sum of first two terms
• the second term
• 3rd term
• 10th term and
• nth term

Solution :

Sum of n terms is given as Sₙ = 5n - n²

➣ To find the first term, put n = 1 in Sₙ = 5n - n² , since sum of 1 term is itself the first term.

 S₁ = a₁ = 5(1) - 1²

 a₁ = 5 - 1

 a₁ = 4

∴ first term = 4

➣ To find the Sum of first two terms, put n = 2 in Sₙ = 5n - n² ,

 S₂ = 5(2) - 2²

 S₂ = 10 - 4

 S₂ = 6

∴ Sum of first 2 terms = 6

➣ Sum of first two terms is the sum of first term and second term.

We know that values of sum of first two terms and first term, substitute.

  S₂ = a₁ + a₂

  6 = 4 + a₂

   a₂ = 6 - 4

   a₂ = 2

∴ second term = 2

⮕ The nth term of an A.P. is given by

    $\underline{\boxed{\bf a_n=a_1+(n-1)d}}$

 a₁ denotes first term

 d denotes common difference

Common difference is the difference between a term and it's preceding term.

d = a₂ - a₁ = 2 - 4 = -2

➣ To find the 3rd term, put the values of a₁ , d and n in the above formula.

  a₃ = 4 + (3 - 1)(-2)

  a₃ = 4 + 2(-2)

  a₃ = 4 - 4

  a₃ = 0

∴ the 3rd term = 0

➣ To find the 10th term, put the values of a₁ , d and n in the above formula.

  a₁₀ = 4 + (10 - 1)(-2)

  a₁₀ = 4 + 9(-2)

  a₁₀ = 4 - 18

  a₁₀ = -14

∴ the 10th term = -14

➣ To find the nth term, put the values of a₁ , d in the formula

 aₙ = 4 + (n - 1)(-2)

 aₙ = 4 - 2n + 2

 aₙ = 6 - 2n

Answer 2

$\huge\mathfrak{☯~Answer}$

• First term = 4
• Sum of first two terms = 6
• Second term = 2
• 3rd term = 0
• 10th term = -14
• nth term = 6 - 2n

$\large\mathfrak{\color{green}{\underline{Step-by-step~ explanation}}}$

$\sf{\blue{Given:}}$

• The sum of the first n terms of an AP is Sₙ = 5n - n²

$\sf{\blue{To ~find :}}$

• The first term
• The sum of first two terms
• The second term
• 3rd term
• 10th term
• nth term

$\huge\mathfrak{\red{Solution:}}$

Sum of n terms is given as Sₙ = 5n - n²

⬗ To find the first term, put n = 1 in Sₙ = 5n - n² , since sum of 1 term is itself the first term.

 S₁ = a₁ = 5(1) - 1²

 a₁ = 5 - 1

 a₁ = 4

∴ First term = 4

______________________

⬗ To find the Sum of first two terms, put n = 2 in Sₙ = 5n - n² ,

 S₂ = 5(2) - 2²

 S₂ = 10 - 4

 S₂ = 6

∴ Sum of first 2 terms = 6

______________________

⬗ Sum of first two terms is the sum of first term and second term.

We know that values of sum of first two terms and first term, substitute.

  S₂ = a₁ + a₂

  6 = 4 + a₂

   a₂ = 6 - 4

   a₂ = 2

∴ Second term = 2

______________________

☆ The nth term of an A.P. is given by

${\purple{\boxed{\bf a_n=a_1+(n-1)d}}}$

•  a₁ denotes first term
•  d denotes common difference

Common difference is the difference between a term and it's preceding term.

d = a₂ - a₁ = 2 - 4 = -2

______________________

⬗ To find the 3rd term, put the values of a₁ , d and n in the above formula.

  a₃ = 4 + (3 - 1)(-2)

  a₃ = 4 + 2(-2)

  a₃ = 4 - 4

  a₃ = 0

∴ The 3rd term = 0

______________________

⬗ To find the 10th term, put the values of a₁ , d and n in the above formula.

  a₁₀ = 4 + (10 - 1)(-2)

  a₁₀ = 4 + 9(-2)

  a₁₀ = 4 - 18

  a₁₀ = -14

∴ The 10th term = -14

______________________

⬗ To find the nth term, put the values of a₁ , d in the formula

 aₙ = 4 + (n - 1)(-2)

 aₙ = 4 - 2n + 2

 aₙ = 6 - 2n

∴ The nth term = 6 - 2n

______________________