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Find the 9 th term of the arithmetic sequence 5x+8,-2x+13,-9x+18,dots Answer:

Find the 99 th term of the arithmetic sequence 5x+8,−2x+13,−9x+18,… 5 x+8,-2 x+13,-9 x+18, \ldots \newlineAnswer:

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Q. Find the 99 th term of the arithmetic sequence 5x+8,−2x+13,−9x+18,… 5 x+8,-2 x+13,-9 x+18, \ldots \newlineAnswer:
  1. Define Common Difference: To find the 9th9^{th} term of an arithmetic sequence, we need to determine the common difference between consecutive terms and then use the formula for the nthn^{th} term of an arithmetic sequence, which is an=a1+(n−1)da_n = a_1 + (n - 1)d, where ana_n is the nthn^{th} term, a1a_1 is the first term, nn is the term number, and dd is the common difference.
  2. Calculate Common Difference: First, let's find the common difference dd by subtracting the first term from the second term.\newlined=(−2x+13)−(5x+8)d = (-2x + 13) - (5x + 8)\newlined=−2x+13−5x−8d = -2x + 13 - 5x - 8\newlined=−7x+5d = -7x + 5
  3. Verify Consistency: Now, let's verify the common difference by subtracting the second term from the third term to ensure it is consistent.\newlined=(−9x+18)−(−2x+13)d = (-9x + 18) - (-2x + 13)\newlined=−9x+18+2x−13d = -9x + 18 + 2x - 13\newlined=−7x+5d = -7x + 5\newlineSince we got the same common difference, we can confirm that the sequence is arithmetic and the common difference is correct.
  4. Find 99th Term: Next, we use the formula for the nth term of an arithmetic sequence to find the 99th term a9a_9.a9=a1+(9−1)da_9 = a_1 + (9 - 1)da9=(5x+8)+8(−7x+5)a_9 = (5x + 8) + 8(-7x + 5)
  5. Simplify Expression: Now, let's simplify the expression for the 9th9^{\text{th}} term.a9=5x+8+8(−7x)+8(5)a_9 = 5x + 8 + 8(-7x) + 8(5)a9=5x+8−56x+40a_9 = 5x + 8 - 56x + 40a9=−51x+48a_9 = -51x + 48

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