the sequence of positive whole numbers is given such that A1<A2 and An+2=An+1+an for n>=1.if a7=120.find A8.
Answers
Answer:
Step-by-step explanation:
Given : the sequence of positive whole numbers is given such that a₁ <a₂ and aₙ₊₂=aₙ₊₁ +aₙ for n≥1. if a₇=120
To find : a₈
Solution:
let say a₁ = a and a₂ = a + k ∵ a₁ < a₂
where a , k are positive whole numbers
a₃ = a₂ + a₁ = a + k + a = 2a + k
a₄ = a₃ + a₂ = 2a+k + a + k = 3a + 2k
a₅ = a₄ + a₃ = 3a + 2k + 2a + k = 5a + 3k
a₆ = a₅ + a₄ = 5a + 3k + 3a + 2k = 8a + 5k
a₇ = a₆ + a₅ = 8a + 5k + 5a + 3k = 13a + 8k
a₈ = a₇ + a₆ = 13a + 8k + 8a + 5k = 21a + 13k
a₇ = 120
a₇ = 13a + 8k
Equating both :
13a + 8k = 120
5a + 8a + 8k = 8 * 15
=> 5a = 8(15 - a - k )
a = 8 and 5 = 15 - a - k => 5 = 15 - 8 - k => k = 2
k = 2
Hence
a₈ = 21a + 13k = 21(8) + 13(2) = 168 + 26 = 194
a₈ = 194
Series is a₁ = 8 and a₂ = 8 + 2 = 10
8 , 10 , 18 , 28 , 46 , 74 , 120 , 194 and so on
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