3. If the sum of first terms of an AP in 4n-n², what is the second term? what is common common differeme and find an.?
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3. If the sum of first terms of an AP in 4n-n², what is the second term? what is common common differeme and find an.?
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Answer:
the sum of the first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.
Solution:
The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1) d] or Sₙ = n/2 [a + l], and the nth term of an AP is aₙ = a + (n - 1)d
Here, a is the first term, d is the common difference and n is the number of terms and l is the last term.
Given,
Sum of first n terms, Sₙ = 4n - n²
Therefore,
Sum of first term, S₁ = 4 × 1 - 1² = 4 - 1 = 3
Sum of first two terms, S₂ = 4 × 2 - 2² = 8 - 4 = 4
Sum of first three terms, S₃ = 4 × 3 - 3² = 12 - 9 = 3
Second term, a₂ = S₂ - S₁ = 4 - 3 = 1
Third term, a₃ = S₃ - S₂ = 3 - 4 = - 1
Tenth term, a₁₀ = S₁₀ - S₉
= (4 × 10 - 102) - (4 × 9 - 92)
= (40 - 100) - (36 - 81)
= - 60 + 45
= - 15
nth term, aₙ = Sₙ - Sₙ₋₁
= [4n - n²] - [4 (n - 1) - (n - 1)² ]
= 4n - n² - 4n + 4 + (n - 1)²
= 4 - n² + n² - 2n + 1
= 5 - 2n