Question

Given that 2k, 5 and 6 - k are the first three terms in an arithmetic progression , what is the value of common difference​

Answer 1

✬ Common Difference = – 3 ✬

Step-by-step explanation:

Given:

• First three terms of an AP are 2k , 5 and 6 – k respectively.

To Find:

• What is the common difference of AP ?

Solution: Let the common difference be d.

As we know that

• a = first term

• a + d = second term

• a + 2d = third term

• d = common difference

Now we have

➟ a = 2kㅤㅤㅤㅤㅤ(first term)

➟ a + d = 5ㅤㅤㅤㅤㅤ(second term)

➟ a + 2d = 6 – kㅤㅤㅤㅤㅤ(third term)

∴ d = second term – first term

➮ d = 5 – 2kㅤㅤㅤㅤ(eqⁿ 1)

Again,

➮ d = third term – second term

➮ d = 6 – k – 5

➮ d = 1 – kㅤㅤㅤㅤ(eqⁿ 2)

According to the question

$\implies{\rm }$ d = d

$\implies{\rm }$ 5 – 2k = 1 – k

$\implies{\rm }$ 5 – 1 = – k + 2k

$\implies{\rm }$ 4 = k

Let's put the value of 'k' in all the terms.

➙ First term = 2 × 4 = 8

➙ Second term = 5

➙ Third term = 6 – 4 = 2

∴ d = second term – first term

➮ d = 5 – 8 = – 3

➮ d = 2 – 5 = – 3

Hence, the common difference of AP is – 3.

Answer 2

Given :-

Given  that 2k, 5 and 6 - k are the first three terms in an arithmetic progression

To Find :-

Common difference

Solution :-

From the given clue

d = 5 - 2k(I)

d = 6 - k - 5

d = 6 - 5 - k

d = 1 - k(1)

Now

Let the terms be a, a + d, a + 2d

ATQ

5 - 2k = 1 - k

2k - k = 5 - 1

k = 4

Now

by putting k in equation 2

d = 1 - 4

d = -3

The terms

First term = 2k = 2(4) = 8

Second term = 5 = 5

Third term = 6 - k = 6 - 4 = 2