Question

find the value of k and determine the common difference of the arithmetic sequence 5k -3 k + 2 3k -11​

Answer 1

$We \: know \:that ,$

$\pink{ If \: a,b \: and \: c \: are \: in \:a.p \:then} \\\pink{b-a = c - b}$

$Given \: (5k-3), (k+2) \: and \: (3k-11) \:are \\in \:A.P$

$Here, a = 5k-3, \: b = k+2 \: and \: c = 3k-11$

$\implies (k+2)-(5k-3) = (3k-11)-(k+2)$

$\implies k+2-5k+3 = 3k-11-k-2$

$\implies -4k + 5 = 2k - 13$

$\implies -4k-2k = -13 - 5$

$\implies -6k = -18$

$\implies k = \frac{-18}{-6}$

$\implies k = 3$

$Common \: difference (d) = a_{2} - a_{1}$

$= (k+2) - (5k-3)$

$= k +2 - 5k + 3$

$= -4k +5$

$= (-4) \times 3 + 5$

$= -12+5\\= -7$

Therefore.,

$\red{ Value \: of \: k} \green { = 3 }$

$\red{ Common \: difference (d)} \green { = -7}$

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Answer 2

Hope it helps a lot..............