The algebra of an arithmetic sequence is 4n -
1 . a) What is the common difference? b) what is the remainder when each term is divided by the common difference?
c) verify whether 2018 is a term of this sequence?
Answers
EXPLANATION.
Algebraic of an arithmetic sequence : 4n - 1.
As we know that,
⇒ Tₙ = 4n - 1.
Put the value of n = 1 in the equation, we get.
⇒ T₁ = 4(1) - 1.
⇒ T₁ = 3.
Put the value of n = 2 in the equation, we get.
⇒ T₂ = 4(2) - 1.
⇒ T₂ = 7.
Put the value of n = 3 in the equation, we get.
⇒ T₃ = 4(3) - 1.
⇒ T₃ = 11.
Put the value of n = 4 in the equation, we get.
⇒ T₄ = 4(4) - 1.
⇒ T₄ = 15.
Arithmetic sequence : 3, 7, 11, 15, . . . . .
First term = a = 3.
Common difference = d = b - a = 7 - 3 = 4.
To find :
(1) What is the common difference.
Common difference = d = 4.
(2) What is the remainder when each term is divided by the common difference.
⇒ T₁ = 3.
⇒ 3(1) - 0.
⇒ T₂ = 7.
⇒ 4(1) + 3.
⇒ T₃ = 11.
⇒ 4(2) + 3.
⇒ T₄ = 15.
⇒ 4(3) + 3.
Remainder : 0, 3, 3, 3, . . . . .
(3) Verify whether 2018 is a term of the sequence.
As we know that,
General terms of an ap.
⇒ Tₙ = a + (n - 1)d.
Using this formula in this question, we get.
First term = a = 3.
Common difference = d = 4.
⇒ Tₙ = 2018.
⇒ 2018 = 3 + (n - 1)(4).
⇒ 2018 = 3 + 4n - 4.
⇒ 2018 = 4n - 1.
⇒ 2018 + 1 = 4n.
⇒ 2019 = 4n.
⇒ n = 2019/4.
No, 2018 is not a term of the sequence.