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Find the 14 th term of the arithmetic sequence -2x-3,-6x-9,-10 x-15,dots Answer:

Find the 1414 th term of the arithmetic sequence −2x−3,−6x−9,−10x−15,… -2 x-3,-6 x-9,-10 x-15, \ldots \newlineAnswer:

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Q. Find the 1414 th term of the arithmetic sequence −2x−3,−6x−9,−10x−15,… -2 x-3,-6 x-9,-10 x-15, \ldots \newlineAnswer:
  1. Verify Common Difference: Verify that the common difference remains consistent by subtracting the second term from the third term.\newline−10x−15−(−6x−9)=−10x−15+6x+9=−4x−6-10x-15 - (-6x-9) = -10x-15 + 6x + 9 = -4x - 6\newlineThis confirms that the common difference is indeed −4x−6-4x - 6.
  2. Use Arithmetic Sequence Formula: Use the formula for the nnth term of an arithmetic sequence: an=a1+(n−1)da_n = a_1 + (n-1)d, where ana_n is the nnth term, a1a_1 is the first term, and dd is the common difference.\newlineHere, a1=−2x−3a_1 = -2x-3, d=−4x−6d = -4x - 6, and n=14n = 14.
  3. Substitute Values: Substitute the values into the formula to find the 14th14^{th} term.a14=−2x−3+(14−1)(−4x−6)a_{14} = -2x-3 + (14-1)(-4x - 6)a14=−2x−3+13(−4x−6)a_{14} = -2x-3 + 13(-4x - 6)
  4. Simplify Expression: Simplify the expression by distributing 1313 into (−4x−6)(-4x - 6). \newlinea14=−2x−3+(−52x−78)a_{14} = -2x-3 + (-52x - 78)\newlinea14=−2x−3−52x−78a_{14} = -2x-3 - 52x - 78
  5. Combine Like Terms: Combine like terms to get the final expression for the 14th14^{\text{th}} term.a14=−54x−81a_{14} = -54x - 81

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