Question

If the first 8 and 10 term of an A.P are 64 and 361 respectively, find the common difference and the sun of n term of the series

Answer 1

t8=64
t10=361

tn=a+(n-1)d
t8=a+(8-1)d
64=a+7d .…..1

t10=a+(10-1)d
361=a+9d. ......2

substract equation 1 from 2
a + 9d =361
_a + 7d = 64
2d. = 297
d = 148.5

Answer 2

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.