A-Level Mathematics May/June 2024 Q8(a): The first three terms of an arithmetic progression are 25, 4p – 1 and 13-p, where p is…

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  For an arithmetic progression, the difference between consecutive terms is constant. Let the terms be $u_1, u_2, u_3$.

  $u_1 = 25$ $u_2 = 4p - 1$ $u_3 = 13 - p$

  The common difference $d$ is given by $u_2 - u_1 = u_3 - u_2$. This can be rearranged to $2u_2 = u_1 + u_3$.

  Substituting the given terms: $2(4p - 1) = 25 + (13 - p)$ $8p - 2 = 38 - p$ $9p = 40$ $p = \frac{40}{9}$

  Now, we find the common difference, $d$. The first term is $a = 25$. $d = u_2 - u_1 = (4p - 1) - 25$ $d = 4p - 26$

  Substitute the value of $p$: $d = 4\left(\frac{40}{9}\right) - 26$ $d = \frac{160}{9} - \frac{234}{9}$ $d = -\frac{74}{9}$

  Finally, we find the tenth term, $u_{10}$, using the formula $u_n = a + (n-1)d$. $u_{10} = a + 9d$ $u_{10} = 25 + 9\left(-\frac{74}{9}\right)$ $u_{10} = 25 - 74$ $u_{10} = -49$

  The value of the tenth term is $-49$.

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  How the marks are awarded

  • *M1 — The method mark is awarded for setting up a correct equation to find the value of $p$ by using the property of an arithmetic progression, i.e. $2u_2 = u_1 + u_3$, which leads to the equation $2(4p-1) = 25 + (13-p)$.
  • A1 — The accuracy mark is for correctly solving the linear equation to find the value of $p = \frac{40}{9}$.
  • DM1 — This dependent method mark is for using their calculated value of $p$ to find the common difference, $d$. The model answer shows this with $d = 4(\frac{40}{9}) - 26 = -\frac{74}{9}$.
  • A1 — The final accuracy mark is for correctly calculating the 10th term using the formula $u_{10} = a + 9d$ with the first term $a=25$ and their calculated common difference, leading to the final answer of $-49$.

  Common mistakes

  • Setting up the initial equation incorrectly, for example using the property of a geometric progression ($u_2^2 = u_1 u_3$) or an incorrect arithmetic relationship like $u_1 + u_2 = u_3$.
  • Making an algebraic error when solving for $p$, such as expanding $2(4p-1)$ to $8p-1$ or incorrectly collecting terms, leading to the wrong value for $p$ and all subsequent calculations.
  • Using the wrong formula for the 10th term, a common error being to use $u_{10} = a + 10d$ instead of the correct $u_{10} = a + 9d$.
  • Calculating the value of $p$ correctly but then stopping, failing to answer the actual question which asks for the value of the tenth term.

  Examiner tip: For any sequence with unknown algebraic terms, use the fundamental definition of that sequence (e.g., a constant common difference for an AP) to create an equation and solve for the unknown variable.

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