If the sums of the first 8 and 19 terms of an AP are 64 and 361 respectively, find the common difference and the sum of n terms of the series.
Answers
Here is your answer:
ATQ.
S8 = 64 = 4(2a+7d)
64/4 = 2a+7d
16 = 2a+7d------------------(1)
S19 = 361 = 19/2(2a+18d)
361*2/19 = 2a+18d
38 = 2(a+9d)
19 = a+9d = A10 --------------------(2)
From 2
a = 19-9d
Putting value of a in eq-(1)
16 = 2(19-9d) + 7d
16 = 38 - 18d + 7d
-22 = -11d
d = 22/11
d = 2
Therefore,
19 = a+9*2
a = 19-18
a = 1
Sum of n terms = Sn
= n/2(2*1+(n-1)2)
= n²
MARK IT AS THE BRAINLIEST PLZZZZ..............
Answer:
• Sum of n terms in AP :
Sn = (n/2)[2a + (n- 1)d]
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⇒ S₈ = 64
⇒ 8/2 × (2a + 7d) = 64
⇒ 4 × (2a + 7d) = 64
⇒ 2a + 7d = 16 — eq. ( I )
⇒ S₁₉ = 361
⇒ 19/2 × (2a + 18d) = 361
⇒ 19 × (a + 9d) = 361
⇒ a + 9d = 19 — eq. ( II )
• Multiplying eq.( II ) by 2 & Subtracting from eq.( I ) from eq.( II ) :
↠ 2a + 18d - 2a - 7d = 38 - 16
↠ 11d = 22
↠ d = 2
• Substitute d value in eq. ( II ) :
⇒ a + 18 = 19
⇒ a = 19 - 18
⇒ a = 1
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⋆ Sum of nth terms of the AP :
↠ Sn = n/2 [2a + (n - 1)d]
↠ Sn = n/2 × [2 × 1 + (n - 1) × 2]
↠ Sn = n/2 × [2 + 2n - 2]
↠ Sn = n/2 × 2n
↠ Sn = n × n
↠ Sn = n²
∴ Sum of nth terms of the AP is n².