Arithmetic Progressions

Question

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9^th terms.

Solution

Let a_1,a₂ be the first terms and d₁, d₂ the common difference of the two given A.P.'s

Then, sum of their n terms is given by

$S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d_{1}] and S_{n} = \frac{n}{2} [2 a_{2} + (n - 1) d_{2}] ∴ \frac{s_{n}}{s_{n}} = \frac{\frac{n}{2} [2 a_{1} + (n - 1) d_{1}]}{\frac{n}{2} [2 a_{2} + (n - 1) d_{2}]} = \frac{[2 a_{1} + (n - 1) d_{1}]}{[2 a_{2} + (n - 1) d_{2}]}$

It is given that,

$\frac{s_{n}}{s_{n}} = \frac{7 n + 1}{4 n + 27} \Rightarrow \frac{2 a_{1} + (n - 1) d_{1}}{2 a_{2} + (n - 1) d_{2}} = \frac{7 n + 1}{4 n + 27} . . . . . . . . . . (i)$

In order to find the ratio of the m^th terms of the two given A,P.'s,

We replace n by [ 2m-1] in equation (i)

$\frac{2 a_{1} + (17 - 1) d_{1}}{2 a_{2} + (17 - 1) d_{2}} = \frac{7 x 17 + 1}{4 x 17 + 27} \Rightarrow \frac{2 a_{1} + 16 d_{1}}{2 a_{2} + 16 d_{2}} = \frac{24}{19} \Rightarrow \frac{a_{1} + 8 d_{1}}{a_{2} + 8 d_{2}} = \frac{24}{19}$

Thus, the ratio of their 9^th terms is 24:19.

Some More Questions From Arithmetic Progressions Chapter

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?

The cost of digging a well for the first metre is Rs. 150 and rises by Rs. 50 for each succeeding metre.

For Daily Free Study Material Join wiredfaculty

Arithmetic Progressions

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9^th terms.

Some More Questions From Arithmetic Progressions Chapter

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?

The cost of digging a well for the first metre is Rs. 150 and rises by Rs. 50 for each succeeding metre.

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a – 10, d = 10

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = –2, d = 0

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = 4, d = – 3

Write first four terms of the AP, when the first term a and the common difference d are given as follows:

a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference
3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference
– 5, – 1, 3, 7, . . .

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Arithmetic Progressions

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.

Some More Questions From Arithmetic Progressions Chapter

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?The cost of digging a well for the first metre is Rs. 150 and rises by Rs. 50 for each succeeding metre.

Write first four terms of the AP, when the first term a and the common difference d are given as follows:a – 10, d = 10

Write first four terms of the AP, when the first term a and the common difference d are given as follows:a = –2, d = 0

Write first four terms of the AP, when the first term a and the common difference d are given as follows:a = 4, d = – 3

Write first four terms of the AP, when the first term a and the common difference d are given as follows:a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference– 5, – 1, 3, 7, . . .

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9^th terms.

In which of the following situational, does the list of numbers involved make an arithmetic progression, and why?

The cost of digging a well for the first metre is Rs. 150 and rises by Rs. 50 for each succeeding metre.

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a – 10, d = 10

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = –2, d = 0

Write first four terms of the AP, when the first term a and the common difference d are given as follows:
a = 4, d = – 3

Write first four terms of the AP, when the first term a and the common difference d are given as follows:

a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference
3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference
– 5, – 1, 3, 7, . . .