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In an arithmetic sequence, the first term, 
a_(1), is equal to 5 , and the third term, 
a_(3), is equal to 9 . Which number represents the common difference of the arithmetic sequence?

d=2

d=3

d=4

d=5

In an arithmetic sequence, the first term, a1 a_{1} , is equal to 55 , and the third term, a3 a_{3} , is equal to 99 . Which number represents the common difference of the arithmetic sequence?\newlined=2 d=2 \newlined=3 d=3 \newlined=4 d=4 \newlined=5 d=5

Full solution

Q. In an arithmetic sequence, the first term, a1 a_{1} , is equal to 55 , and the third term, a3 a_{3} , is equal to 99 . Which number represents the common difference of the arithmetic sequence?\newlined=2 d=2 \newlined=3 d=3 \newlined=4 d=4 \newlined=5 d=5
  1. Identify Given Terms: Identify the given terms in the arithmetic sequence.\newlineWe are given the first term a1a_1 and the third term a3a_3 of the arithmetic sequence.\newlinea1=5a_1 = 5\newlinea3=9a_3 = 9
  2. Use Formula for nth Term: Use the formula for the nth term of an arithmetic sequence to find the common difference.\newlineThe formula for the nth term of an arithmetic sequence is:\newlinean=a1+(n1)da_n = a_1 + (n - 1)d\newlinewhere ana_n is the nth term, a1a_1 is the first term, dd is the common difference, and nn is the term number.
  3. Plug Values and Find dd: Plug the values of a1a_1 and a3a_3 into the formula to find dd. We know that a3=a1+2da_3 = a_1 + 2d (since the third term is two terms away from the first term). So, 9=5+2d9 = 5 + 2d
  4. Solve for d: Solve for d.\newlineSubtract 55 from both sides of the equation:\newline95=2d9 - 5 = 2d\newline4=2d4 = 2d\newlineNow, divide both sides by 22 to find d:\newlined=42d = \frac{4}{2}\newlined=2d = 2

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