A manufacture of TV sets produces 600 TV sets in the third year and 700 TV sets in seventh year. Assuming that the production increases, uniformly by a fixed number every year, find
(i) the production in the 1^st year
(ii) the production in the 10^th year
(iii) the total production in first 7 years.

Given, a₃ = 600 and a₇ = 700

Let the first term be a and common difference be d.
Since, nth term an AP is given by :
a_n = a + (n - 1)d

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

⇒ a₃ = a + (3 - 1)d
⇒ 600 = a + 2d … (i)

⇒ a₇ = a + (7 - 1)d
⇒ 700 = a + 6d …(ii)

On solving eq. (i) and (ii), we get :
a = 550 and d =25

∴ production in first year a₁ = a 550
and production in tenth year a₁₀
= a + (10 - 1)d
= 550 + 9 × 25
= 550 + 225 = 775
∴ a₁₀ = 775

Since the sum of n terms is
S_{n =}
Total production in first seven year S₇
S_{7 =}
S₇ = 3.5 × [ (2× 550) + (6 × 25)]
S₇ = 3.5 × [ 1100 + 150]
S₇ = 3.5 × 1250
S₇ = 4375

Hence, (i) the production in the 1^st year is 550
(ii) the production in the 10^th year is 775
(iii) the total production in first 7 years is 4375.