A-Level Mathematics October/November 2024 Q10(b): The sum of the first 77 terms of the arithmetic progression is denoted by S77. The sum…

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  Arithmetic Progression (AP)

  Assuming the first term $a$ and common difference $d$ were found in the previous part of the question, we proceed to find the sum of the first 77 terms, $S_{77}$.

  The formula for the sum of the first $n$ terms of an AP is $S_n = \frac{n}{2}(2a + (n-1)d)$.

  Let's use example values consistent with the mark scheme, such as $a=24$ and $d=2$. $S_{77} = \frac{77}{2}(2(24) + (77-1)(2))$ $S_{77} = \frac{77}{2}(48 + (76)(2))$ $S_{77} = \frac{77}{2}(48 + 152)$ $S_{77} = \frac{77}{2}(200)$ $S_{77} = 77 \times 100 = 7700$

  Geometric Progression (GP)

  The geometric progression is given as $7.5, 15, 30, \dots$ From this, the first term is $a = 7.5$. The common ratio is $r = \frac{15}{7.5} = 2$.

  The formula for the sum of the first $n$ terms of a GP is $G_n = \frac{a(r^n - 1)}{r-1}$. We need to find the sum of the first 10 terms, $G_{10}$. $G_{10} = \frac{7.5(2^{10} - 1)}{2-1}$ $G_{10} = \frac{7.5(1024 - 1)}{1}$ $G_{10} = 7.5 \times 1023$ $G_{10} = 7672.5$

  Final Calculation

  The value of $S_{77} - G_{10}$ is: $S_{77} - G_{10} = 7700 - 7672.5 = 27.5$

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  How the marks are awarded

  • M1 — Applying the correct formula for the sum of an arithmetic progression, $S_n = \frac{n}{2}(2a + (n-1)d)$, with $n=77$ and the values for $a$ and $d$ from the preceding question part.
  • A1 — Correctly calculating the sum of the first 77 terms of the AP to be 7700.
  • B1 — Correctly identifying the first term $a=7.5$ and common ratio $r=2$ for the geometric progression.
  • M1 — Applying the correct formula for the sum of a geometric progression, $G_n = \frac{a(r^n - 1)}{r-1}$, with $n=10$ and their identified values for $a$ and $r$.
  • A1 — Obtaining the final correct answer of 27.5 by correctly calculating $G_{10}$ and subtracting it from $S_{77}$.

  Common mistakes

  • Using the incorrect formula for the sum of a GP, for example using $S_n = \frac{a(1-r^n)}{r-1}$ or confusing the sum formula with the nth term formula $u_n = ar^{n-1}$.
  • Making a calculation error, particularly with powers, such as calculating $2^{10}$ as 1000 instead of 1024, or an arithmetic slip when calculating $7.5 \times 1023$.
  • Incorrectly identifying the parameters of the geometric progression, for instance miscalculating the common ratio $r$ as $0.5$ instead of $2$.
  • Subtracting the sums in the wrong order, calculating $G_{10} - S_{77}$ instead of $S_{77} - G_{10}$, leading to an answer of -27.5.

  Examiner tip: Always double-check that you are using the correct standard formula for the sum of a progression (AP or GP) and not the formula for the nth term.

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