A-Level Mathematics May/June 2024 Q7(b): Find the sum of all the terms of the arithmetic progression whose values are between 25…

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  The arithmetic progression is defined by $u_n = a + (n-1)d$. From the context of the question, we use $a = 1.5$ and $d = 2.5$. The general term is $u_n = 1.5 + (n-1)2.5$.

  We need to find the terms whose values are strictly between 25 and 100.

  First, find the first term greater than 25: $1.5 + (n-1)2.5 > 25$ $(n-1)2.5 > 23.5$ $n-1 > \frac{23.5}{2.5}$ $n-1 > 9.4$ $n > 10.4$ Since $n$ must be an integer, the first term in the required range is the 11th term ($n=11$). Value of the first term to be summed: $u_{11} = 1.5 + (11-1)2.5 = 1.5 + 10(2.5) = 26.5$.

  Next, find the last term less than 100: $1.5 + (n-1)2.5 < 100$ $(n-1)2.5 < 98.5$ $n-1 < \frac{98.5}{2.5}$ $n-1 < 39.4$ $n < 40.4$ Since $n$ must be an integer, the last term in the required range is the 40th term ($n=40$). Value of the last term to be summed: $u_{40} = 1.5 + (40-1)2.5 = 1.5 + 39(2.5) = 99$.

  We need to find the sum of the terms from $n=11$ to $n=40$. The number of terms to be summed is $N = 40 - 11 + 1 = 30$. This is an arithmetic progression with: First term, $a' = 26.5$ Last term, $l' = 99$ Number of terms, $N = 30$

  Using the formula for the sum of an arithmetic progression, $S_N = \frac{N}{2}(a' + l')$: $S_{30} = \frac{30}{2}(26.5 + 99)$ $S_{30} = 15(125.5)$ $S_{30} = 1882.5$

  The sum of all the terms is 1882.5.

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  How the marks are awarded

  • M1 — The M1 is awarded for attempting to find the term numbers for the start or end of the range. In the model answer, this is shown by setting up the inequalities $1.5 + (n-1)2.5 > 25$ or $1.5 + (n-1)2.5 < 100$ and proceeding to solve for n.
  • A1 — The A1 is awarded for correctly identifying the first term in the sequence to be summed. This is achieved by finding that $n>10.4$, leading to the 11th term, and stating its value is $26.5$.
  • A1 — This A1 is awarded for correctly identifying the last term. This is shown by finding $n<40.4$, leading to the 40th term, and stating its value is $99$.
  • DM1 — This dependent method mark is awarded for using a correct sum formula with the candidate's derived values. The model answer uses $S_N = \frac{N}{2}(a' + l')$ with the correctly calculated number of terms ($N=30$), first term ($a'=26.5$), and last term ($l'=99$).
  • A1 — The final A1 is awarded for the correct final answer of $1882.5$.

  Common mistakes

  • An 'off-by-one' error when calculating the number of terms to be summed, for example using $40 - 11 = 29$ instead of the correct $40 - 11 + 1 = 30$.
  • Incorrectly rounding the results of the inequalities. For example, from $n > 10.4$, incorrectly concluding that the 10th term is the first one in the range.
  • Using the original first term of the progression ($a=1.5$) in the sum formula instead of the first term that is actually within the specified range ($a'=26.5$).
  • When using the alternative method of subtracting sums ($S_{40} - S_{10}$), incorrectly calculating $S_{40} - S_{11}$, which omits the 11th term from the sum.

  Examiner tip: When summing a sub-section of a sequence, always clearly identify the first term, the last term, and the total number of terms in the new sequence you are summing.

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