Question

The algebraic of an arithmetic sequence is xn = 3n+2.

a) Write it's first term and common difference.

b) What is the remainder on dividing each terms of this sequence by 3.

c) Is 100 , a term in this sequence.Why.


please help me to write this answer step by step​

Answer 1

EXPLANATION.

Algebraic of an arithmetic sequence.

⇒ xₙ = 3n + 2.

As we know that,

Put the value of n = 1 in the equation, we get.

⇒ x₁ = 3(1) + 2.

⇒ x₁ = 5.

Put the value of n = 2 in the equation, we get.

⇒ x₂ = 3(2) + 2.

⇒ x₂ = 8.

Put the value of n = 3 in the equation, we get.

⇒ x₃ = 3(3) + 2.

⇒ x₃ = 11.

Put the value of n = 4 in the equation, we get.

⇒ x₄ = 3(4) + 2.

⇒ x₄ = 14.

Series = 5, 8, 11, 14 . . . . .

First term = a = 5.

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

Common difference = d = 8 - 5 = 3.

Dividing each term of this sequence by 3, we get.

Series = (5/3), (8/3), (11/3), (14/3), . . . . .

We get Remainder = 2.

Is, 100 a term in this sequence.

As we know that,

Formula of :

General term of an A.P.

⇒ Aₙ = a + (n - 1)d.

Let we assume that,

⇒ Aₙ = 100.

100 = 5 + (n - 1)3.

⇒ 100 = 5 + 3n - 3.

⇒ 100 = 2 + 3n.

⇒ 100 - 2 = 3n.

⇒ 98 = 3n.

⇒ n ≠ 98/3.

There is not an exact value of n.

Hence, 100 is not in the term.

                                                                                                                     

MORE INFORMATION.

Supposition of terms in an 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

Given :-

The algebraic of an arithmetic sequence is :

${\bold{\sf{x{\small_{n}} = 3{\small_{n}} + 2}}}$

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To Find :-

1.) Write it's first term and common difference.

2.) What is the remainder on dividing each terms of this sequence by 3.

3.) Is 100 , a term in this sequence.Why ?

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Solution :-

1.)

${\bold{\sf{x{\small_{n}} = 3{\small_{n}} + 2}}}$

➢ First putting the value of n as 1 :

• ➻ xn = 3n + 2
• ➻ x 1 = 3(1) + 2
• ➻ x1 = 3 + 2 = 5

➢ Putting the value of n as 2 :

• ➻ x 2 = 3(2) + 2
• ➻ x2 = 6 + 2 = 8

➢ Putting the value of n as 3 :

• ➻ x3 = 3(3) + 2
• ➻x3 = 9 + 2 = 11

➢Putting the value of n as 4 :

• ➻ x4 = 3(4) + 2
• ➻ x4 = 12 + 2 = 14

Series = 5,8,11,14

• First term = 5
• Common difference = b - a = 8 - 5 = 3

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2.)

Let's divide each term by 3 .Hence ,

• ➻ (5/2) , (8/2) ,(11/2) ,(14 /2)

• Remainder = 2

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3.)

➢ Formula Used :

$\large\green\bigstar{\underline{\boxed{\color{maroon}{\sf{A{\small_{(n)} = a + (n - 1)d}}}}}}$

➢Here :

• ➳ An = 100th term = 100
• ➳ a = 1st term = 5
• ➳ n = no. of terms = 3n
• ➳ d = common difference = 3

➢Let's Check :

• ➻ An = a + (n - 1)d
• ➻ A100 = 5(3 - 1) 3
• ➻ 100 = 5 + 3n - 3
• ➻ 100 = 5 - 3 + 3n
• ➻ 100 = 2 + 3n
• ➻ 100 - 2 = 3n
• ➻ 98/3 = n

Hence,

100 is not the term of sequence .

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