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How many terms are there in an AP whose first term and 6th term are -12, and 8 respectively , and sum of all its term is 120 ??
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How many terms are there in an AP whose first term and 6th term are -12, and 8 respectively , and sum of all its term is 120 ??
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Answer:
Step-by-step explanation:
Answer:
12 terms
Step-by-step explanation:
As the question states,
1st term, a = -12
6th term, a₆ = 8
Also, given Sum of all terms, Sₙ = 120
Now,
a₆ = 8
⇒ a + 5d = 8
⇒ -12 + 5d = 8 (Since, 1st term, a = -12)
⇒ 5d = 8 + 12
⇒ 5d = 20
⇒ d = 4
Now,
Sₙ = n/2{2a + (n-1)d}
⇒ 120 = n/2{2a + (n-1)d} (Since, Sₙ = 120)
⇒ 120 = n/2{2 × -12 + (n-1)4} (Since a = -12, d = 4)
⇒ 120 × 2 = n{-24 + 4n - 4}
⇒ 240 = n{-28 + 4n}
⇒ 240 = -28n + 4n²
⇒ 4n² - 28n - 240 = 0
⇒ n² + 7n - 60 = 0
⇒ n² - (12 - 5)n - 60 = 0
⇒ n² - 12n + 5n - 60 = 0
⇒n(n - 12) + 5(n - 12) = 0
⇒ (n - 12)(n + 5) = 0
∴ Either n - 12 = 0 OR n + 5 = 0
∴ Either n = 12 OR n = -5
∴ n = 12 (Since, number of terms cannot be negative)
∴ There are 12 terms in the A.P.