A manufacturer of TV sets produced $600$ sets in the third year and $700$ sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find:
(i) The production in the $1^{\text{st}}$ year
(ii) The total production in first $7$ years
Step-by-Step Solution
Key Concept: $a_3 = a + 2d = 600$, $a_7 = a + 6d = 700 \Rightarrow 4d = 100 \Rightarrow d = 25, a = 550$.<br>(i) $a_1 = 550$ sets.<br>(ii) $S_7 = \dfrac{7}{2}(550 + 700) = \dfrac{7}{2}(1250) = 4375$ sets.
$a + 6d - (a + 2d) = 100 \Rightarrow 4d = 100 \Rightarrow d = 25$. $a = 600 - 50 = 550$. [1.0 Mark]
(i) Production in $1^{\text{st}}$ year $= 550$ sets. [1.0 Mark]
(ii) $S_7 = \dfrac{7}{2}(a_1 + a_7) = \dfrac{7}{2}(550 + 700) = 4375$ sets. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Solving $d = 25, a = 550$: 1.0 Mark
Part (i) $1^{\text{st}}$ year production $= 550$: 1.0 Mark
Part (ii) Total production in 7 years $= 4375$: 1.0 Mark
Correct Answer: