Question

if the sum of the first 8 and 19 terms of an AP are 64 and 361 respectively, find the d and the sum of n terms of the series.

Answer 1

Answer: d=2 and sum of n terms = n²

Step-by-step explanation:

let the first term be a and the common difference be d

∴A.T.Q

8/2[2a+(8-1)d]=64

⇒2a+7d=16          ......(i)

and

19/2[2a+(19-1)d]=361

⇒2a+18d=38

⇒(2a+7d)+11d=38

⇒16+11d=38         [from eq. (i)]

⇒11d=22

⇒d=2

∴d=2

putting d=2 in eq. (i) we get,

2a+14=16

⇒2a=2

⇒a=1

∴a=1

the common difference (d) is 2

and the sum of n terms

=n/2[2a+(n-1)d]

=n/2[2+2n-2]

=n/2*2n

=n²

Answer 2

Answer:

• Sum of n terms in AP :

Sn = (n/2)[2a + (n- 1)d]

───────────────

⇒ 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.