Question

the first three terms of an A.P are b, c and 2b, then find the ratio of b and c​

Answer 1

Given :

b, c and 2b are in Airthemtic progression

To Find :

The ratio of b and c

Theory :

• Arithmetic progression

The sequence $\sf\:x_{1},x_2,x_3.....,x_n$ is called Arithmetic progression.

If $\sf\:x_2-x_1=x_3-x_2=x_n-x_{n-1}$

In general $\sf\:x_n-x_{n-1}=constant$,n€ N .This constant distance is called common difference.

Solution :

We have,

b ,c and 2 b are in Ap

Then the common differences

$\sf\:c-b=2b-c$

$\sf\:-b-2b=-c-c$

$\sf\:-3b=-2c$

Then,

$\sf\:\dfrac{b}{c}=\dfrac{-2}{-3}$

$\sf\implies\:\dfrac{b}{c}=\dfrac{2}{3}$

Therefore , The ratio of b and c is 2:3

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More About topic :

1)Genral term of an ap is given by

$\sf\:a_n=a+(n-1)d$

2) Common difference can can be positive or negative .

3) If there are n terms in AP, then mth term from the end$\sf=(n-m+1)^{th}$ term from the beginning.

•More types of progression

1. Geometric progression
2. Harmonic progression

Answer 2

Answer:

The ratio of b and c is 2:3.

Step-by-step explanation:

If x, y and z are first three terms of an A.P, then

$x+z=2y$

It means the sum of first and third term is twice of second term.

It is given that first three terms of an A.P are b, c and 2b.

$(b)+(2b)=2(c)$

$3b=2c$

Divide both sides by c.

$\frac{3b}{c}=\frac{2c}{c}$

Now divide both sides by 3.

$\frac{b}{c}=\frac{2}{3}$

Therefore, the ratio of b and c is 2:3.