A sequence \(\{a_n\}\) is generated by the recursive formulas \(a_1 = 5\) and \(a_n = a_{n-1} + 5(-1)^n\). Find \(a_{343}\), the 343rd term of the sequence.
Answer (3)
a n = a n − 1 + 5 ( − 1 ) n i f n i s e v e n t h e n : a n = a n − 1 + 5 ( − 1 ) n = a n − 1 + 5 i f n i s o dd t h e n : a n = a n − 1 + 5 ( − 1 ) n = a n − 1 − 5 a 1 = 5 a 2 = 5 + 5 = 10 a 3 = 10 − 5 = 5 a 4 = 5 + 5 = 10 a 5 = 10 − 5 = 5 ⋮ a 343 = 5 ( 343 i s o dd n u mb er )
a 1 = 5 an d a n = a n − 1 + 5 ⋅ ( − 1 ) n ( − 1 ) e v e n n u mb er = 1 an d ( − 1 ) o dd n u mb er = − 1 a 2 = a 1 + 5 ⋅ ( − 1 ) 2 = 5 + 5 = 10 a 3 = a 2 + 5 ⋅ ( − 1 ) 3 = 10 − 5 = 5 = a 1 a 4 = a 3 + 5 ⋅ ( − 1 ) 4 = 5 + 5 = 10 = a 2 i f n → e v e n n u mb er ⇒ a n = 10 i f n → o dd n u mb er ⇒ a n = 5 ⇒ a 343 = 5
To find a 343 in the sequence defined by a 1 = 5 and a n = a n − 1 + 5 ( − 1 ) n , we determine that the sequence has a pattern where odd-indexed terms are 5. Since 343 is odd, a 343 = 5.
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