For what value of \( c \) will \( 2c - 7 \), \( 5c - 9 \), and \( 7c + 2 \) be consecutive terms of an arithmetic sequence?
Answer (2)
2 c − 7 ; 5 c − 9 ; 7 c + 2 − t h e a r i t hm e t i c se q u e n ce t h e n ( 2 c − 7 ) + ( 7 c + 2 ) = 2 ⋅ ( 5 c − 9 ) 2 c − 7 + 7 c + 2 = 10 c − 18 9 c − 5 = 10 c − 18 / + 5 / − 10 c 9 c − 10 c = − 18 + 5 − c = − 13 → c = 13 − an s w er
a 1 ; a 2 ; a 3 − t h e a r i t hm e t i c se q u e n ce a 2 = a 1 + r a 3 = a 2 + r = a 1 + r + r = a 1 + 2 r a 1 + a 3 = a 1 + a 1 + 2 r = 2 a 1 + 2 r = 2 ( a 1 + r ) = 2 a 2 a 1 + a 3 = 2 a 2
The value of c for which the expressions 2 c − 7 , 5 c − 9 , and 7 c + 2 are consecutive terms of an arithmetic sequence is c = 13 . This is determined by setting up the equation based on the properties of an arithmetic sequence and solving for c .
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