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Find the 51 st term of the arithmetic sequence 
21,6,-9,dots
Answer:

Find the 5151 st term of the arithmetic sequence 21,6,9, 21,6,-9, \ldots \newlineAnswer:

Full solution

Q. Find the 5151 st term of the arithmetic sequence 21,6,9, 21,6,-9, \ldots \newlineAnswer:
  1. Given values: To find the 51st51^{\text{st}} term of an arithmetic sequence, we need to know the first term (a1a_1) and the common difference (dd). The first term a1a_1 is given as 2121.
  2. Calculate common difference: The common difference dd can be found by subtracting the second term from the first term. So, d=621d = 6 - 21.
  3. Use nth term formula: Calculating the common difference, we get d=15d = -15.
  4. Substitute values: The formula to find the nnth term of an arithmetic sequence is an=a1+(n1)da_n = a_1 + (n - 1)d. We will use this formula to find the 5151st term (a51a_{51}).
  5. Calculate (n1)(n-1): Substituting the known values into the formula, we get a51=21+(511)(15)a_{51} = 21 + (51 - 1)(-15).
  6. Multiply by common difference: Calculating the term inside the parentheses first, we have (511)=50(51 - 1) = 50.
  7. Add first term: Now, we multiply 5050 by the common difference, which is 15-15. So, 50×(15)=75050 \times (-15) = -750.
  8. Calculate final result: Finally, we add the first term to the product of the common difference and n1n - 1, which gives us a51=21+(750)a_{51} = 21 + (-750).
  9. Calculate final result: Finally, we add the first term to the product of the common difference and n1n - 1, which gives us a51=21+(750)a_{51} = 21 + (-750).Calculating the sum, we find that a51=21750=729a_{51} = 21 - 750 = -729.

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