Question

The third term of an arithmetic progression is 10 and the seventh term is 34.
Find the first term and the common difference

Answer 1

Given :

• a3 = 10
• a7 = 34

To find :

• Common Difference

Solution :

As we know that,

✯ an = a + (n - 1)d

⇒a3 = a + (3 - 1)d

⇒10 = a + 2d ....(1)

____________________

⇒a7 = a + (7 - 1)d

⇒34 = a + 6d ..... (2)

______________________

(2) - (1)

⇒34 - 10 = a + 6d - (a + 2d)

⇒24 = a + 6d - a - 2d

⇒24 = 4d

⇒d = 24/4

⇒d = 6

$\therefore$ Common Difference is 6

_____________________________

We can also find first term of the AP

Put value of d in (1)

⇒10 = a + 2(6)

⇒10 = a + 12

⇒a = 10 - 12

⇒a = -2

$\therefore$ First term of AP is -2

Answer 2

First term is -2 and common difference is 6 .

Solution

We have 3rd time of AP = 10

7th term of AP = 34

◘ a₃ = 10, ◘ a₇ = 34

We know formula for nth term of AP : -

✒ an = a + (n - 1)d

✒ 10 = a + (3 - 1)d

✒ a + 2d = 10 ...(i)

_____________________

✒ 34 = a + (7 - 1)d

✒ a + 6d = 34 ...(ii)

Now substracting (ii) from (i) : -

✒ a + 6d - a - 2d = 34 - 10

✒ 4d = 24

✒ d = 24/4

✒ d = 6

∴ Common difference of AP = 6 .

Now putting this value in (i) : -

✒ a + 2(6) = 10

✒ a + 12 = 10

✒ a = 10 - 12

✒ a = -2

∴ First term of AP = -2 .