Question

the ratio of the 3rd and 6th term of an arithmetic sequence is 4 : 5. Find the ratio of 7th and 11th terms​

Answer 1

Answer:

The ratio of 7th and 11th terms is 4:5.

Step-by-step explanation:

Given :-

• The ratio of the 3rd and 6th term of an arithmetic sequence is 4:5.

To find :-

• The ratio of 7th and 11th terms.

Solution :-

Formula used :-

${\boxed{\sf{T_n=a+(n-1)d}}}$

• a = 1st term
• d = Common difference

Now find the 3rd term and 6th term of the A.P.

★ $\sf{T_3=a+(3-1)d}$

$\to\sf{T_3=a+2d}$

★ $\sf{T_6=a+(6-1)d}$

$\to\sf{T_6=a+5d}$

According to the question,

(a+2d):(a+5d)=4:5

→ $\sf{\dfrac{a+2d}{a+5d}=\dfrac{4}{5}}$

→5a+10d = 4a+20d

→ 5a-4a = 20d -10d

→ a = 10d...............(i)

Now find the 7th and 11th terms of the A.P.

★ $\sf{T_{7}=a+(7-1)d}$

$\to\sf{T_{7}=a+6d}$

★ $\sf{T_{11}=a+(11-1)d}$

$\to\sf{T_{11}=a+10d}$

Ratio of 7th and 11th term,

= $\sf{T_7:T_{11}}$

= (a+6d):(a+10d)

= $\sf{\dfrac{a+6d}{a+10d}}$

• Put a = 10d from eq (I).

= $\sf{\dfrac{10d+6d}{10d+10d}}$

= $\sf{\dfrac{16d}{20d}}$

= $\sf{\dfrac{4}{5}}$

= 4:5

Therefore the ratio of 7th and 11th terms is 4:5.

Answer 2

We know that,

$\rm S_n=a+(n-1)d$

★Here,

• a is first term of the A.P/arithmetic sequence.
• n is the potential of term.
• d is common difference between the terms.
• $\rm S_n$ is number of term.

__________________________

Therefore,

3rd term of the A.P. will be,

$\rm S_3=a+(3-1)d$

$\rm S_3 = a+2d$

Similarly,

6th term of the A.P. will be,

$\rm S_6 = a+(6-1)d$

$\rm S_6 = a+5d$

_________________________

Now it's given that the ratio of the 3rd term and the 6th term of the A.P. is 4 : 5.

Therefore, we can conclude,

$\rm\implies\dfrac{a+2d}{a+5d} = \dfrac{4}{5}$

$\rm\implies 5(a+2d)=4(a+5d)$

$\rm\implies 5a+10d=4a+20d$

$\rm\implies 5a-4a=20d-10d$

$\bf\implies a=10d$

_______________________________

Now we have to find the ratio of 7th and 11th term.

$\rm \dfrac{S_7}{S_{11}} = \dfrac{a+6d}{a+10d}$

$\implies\rm \dfrac{10d+6d}{10d+10d}\qquad\qquad ...[since, \ a=10d]$

$\implies\rm \dfrac{16d}{20d}$

$\implies\rm \dfrac{4d}{5d}$

$\implies\rm \dfrac{4}{5}$

$\bf \implies 4 : 5$

Hence, the ratio of the 7th and 11th term of the arithmetic sequence is 4 : 5.