Question

The sum of first n terms of arithmetic sequence is n²+4n.
​What is its first term? What is its common difference? What is the algebraic form of this sequence​

Answer 1

ANSWER:

• First term = 5

• Common difference = 2

• Algebraic form = 3 + 2n

GIVEN:

• The sum of first n terms of arithmetic sequence is n² + 4n.

TO FIND:

• First term.

• Common difference.

• Algebraic form of the sequence.

EXPLANATION:

Sₙ = n² + 4n

S₁ = 1² + 4(1)

S₁ = 1 + 4

S₁ = 5

S₂ = 2² + 4(2)

S₂ = 4 + 8

S₂ = 12

S₃ = 3² + 4(3)

S₃ = 9 + 12

S₃ = 21

S₁ => implies the first term.

T₁ = 5

T₂ = S₂ - S₁

T₂ = 12 - 5

T₂ = 7

T₃ = S₃ - S₂

T₃ = 21 - 12

T₃ = 9

Common difference = T₂ - T₁ = T₃ - T₂

T₂ - T₁ = 7 - 5 = 2

T₃ - T₂ = 9 - 7 = 2

Common difference = T₂ - T₁ = T₃ - T₂ = 2

Tₙ = a + (n -1)d

Substitute a = 5 and d = 2

Tₙ = 5 + (n - 1)2

Tₙ = 5 + 2n - 2

Algebraic form = Tₙ = 3 + 2n

VERIFICATION:

Sₙ = n/2(2a + (n - 1)d)

Substitute a = 5 and d = 2

Sₙ = n/2(2(5) + (n - 1)2)

Sₙ = n/2(10 + 2n - 2)

Sₙ = n/2(8 + 2n)

Sₙ = n/2(2(4 + n))

Sₙ = n(4 + n)

Sₙ = 4n + n²

HENCE VERIFIED.

Answer 2

$\rm\huge\blue{\underline{\underline{ Question : }}}$

The sum of first n terms of arithmetic sequence is n²+4n. What is its first term? What is its common difference? What is the algebraic form of this sequence.?

$\rm\huge\blue{\underline{\underline{ Solution : }}}$

★ Given that,

• Sum of first n terms of AP : n² + 4n.

★ To find,

• First term. (a)
• Common difference. (d)
• AP series.

★ Let,

$\sf\red{\implies S_{n} = n^{2} + 4n}$

• Take n = 1

$\bf\:\implies S_{1} = (1)^{2} + 4(1)$

$\bf\:\implies S_{1} = 1 + 4$

$\bf\:\implies S_{1} = 5$

• Take n = 2

$\bf\:\implies S_{2} = (2)^{2} + 4(2)$

$\bf\:\implies S_{2} = 4 + 8$

$\bf\:\implies S_{2} = 12$

• Take n = 3

$\bf\:\implies S_{3} = (3)^{2} + 4(3)$

$\bf\:\implies S_{3} = 9 + 12$

$\bf\:\implies S_{3} = 21$

★ Now,

$\bf\:\implies S_{1} = a = 5$

• [ Remember that, the first term is S1 ]

$\bf\:\implies a_{2} = S_{2} - S_{1} = 12 - 5 = 7$

$\bf\:\implies a_{3} = S_{3} - S_{2}= 21 - 12 = 9$

➡ Now, we got the AP series.

$\bf\green{ \implies AP = 5, 7, 9 .... }$

Common difference (d) = a2 - a1

$\bf\:\implies a_{2} - a_{1}$

$\bf\:\implies 7 - 5$

$\bf\:\implies 2$

Hence,the common difference (d) is 2.

$\underline{\boxed{\bf{\purple{ \therefore First \: term = 5}}}}\:\orange{\bigstar}$

$\underline{\boxed{\bf{\purple{ \therefore The \: common \: difference \: (d) = 2.}}}}\:\orange{\bigstar}$

$\underline{\boxed{\bf{\purple{ \therefore \: AP \:series = 5,7,9....}}}}\:\orange{\bigstar}$

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★ Important formulas in AP :

$\bf\green{\implies a_{n} = a + (n -1)d }$

$\bf\green{\implies S_{n} =\frac{n}{2} [2a + (n -1)d }$

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