sum of 11 terms of an ap is 253 and sumo of 12 terms is 300.a) what is the 6th term? b) what is the 12th term? c) what is the common difference? d) what is the algebric expression
Answers
term 6=23
.
term 12=47
.
.
common difference=4
EXPLANATION.
Sum of first 11 terms of an arithmetic sequence is 253.
Sum of first 12 terms of an arithmetic sequence is 300.
As we know that,
Sum of nth terms of an ap.
⇒ Sₙ = n/2[2a + (n - 1)d].
Using this formula in this question, we get.
Sum of first 11 terms of an arithmetic sequence is 253.
⇒ S₁₁ = 253.
⇒ 11/2[2a + (11 - 1)d] = 253.
⇒ 11/2[2a + 10d] = 253.
⇒ 11[2a + 10d] = 253 x 2.
⇒ 2a + 10d = 23 x 2.
⇒ 2a + 10d = 46. - - - - - (1).
Sum of first 12 terms of an arithmetic sequence is 300.
⇒ S₁₂ = 300.
⇒ 12/2[2a + (12 - 1)d] = 300.
⇒ 6[2a + 11d] = 300.
⇒ 2a + 11d = 50. - - - - - (2).
From equation (1) and equation (2), we get.
Subtracting both the equations, we get.
⇒ 2a + 10d = 46. - - - - - (1).
⇒ 2a + 11d = 50. - - - - - (2).
⇒ - - -
We get,
⇒ 10d - 11d = 46 - 50.
⇒ - d = - 4.
⇒ d = 4.
Put the value of d = 4 in the equation (1), we get.
⇒ 2a + 10d = 46. - - - - - (1).
⇒ 2a + 10(4) = 46.
⇒ 2a + 40 = 46.
⇒ 2a = 46 - 40.
⇒ 2a = 6.
⇒ a = 3.
First term = a = 3.
Common difference = d = 4.
To find :
(a) What is it's 6th term.
As we know that,
General terms of an ap.
⇒ Tₙ = a + (n - 1)d.
Using this formula in this expression, we get.
⇒ T₆ = a + (6 - 1)d.
⇒ T₆ = a + 5d.
Put the values in the equation, we get.
⇒ T₆ = 3 + 5 x 4.
⇒ T₆ = 3 + 20.
⇒ T₆ = 23.
6th term of an arithmetic sequence : T₆ = 23.
(b) What is it's 12th term.
As we know that,
General terms of an ap.
⇒ Tₙ = a + (n - 1)d.
Using this formula in this expression, we get.
⇒ T₁₂ = a + (12 - 1)d.
⇒ T₁₂ = a + 11d.
Put the values in the equation, we get.
⇒ T₁₂ = 3 + 11 x 4.
⇒ T₁₂ = 3 + 44.
⇒ T₁₂ = 47.
12th term of an arithmetic sequence : T₁₂ = 47.
(c) What is the common difference.
Common difference = d = 4.
(d) What is it's algebraic form.
As we know that,
General terms of an ap.
⇒ Tₙ = a + (n - 1)d.
Using this formula in this expression, we get.
⇒ Tₙ = 3 + (n - 1)(4).
⇒ Tₙ = 3 + 4n - 4.
⇒ Tₙ = 4n - 1.
Algebraic form : Tₙ = 4n - 1.