Grade 12 Maths · Number Patterns, Sequences and Series · Find a, d or r from two facts

Find a, d or r from two facts Grade 12

Write two equations from two given terms or sums, for example T_3 = 7 and T_8 = 22, and solve them together to find the first term and the common difference or ratio.

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Question 3.1 · March 2021 · Provincial · 5 marks

3.1 The second term of a geometric sequence is

\frac{5}{128}

and the ninth term is

5

. Determine the value of the common ratio.(5)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More questions: find a, d or r from two facts

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3. Question 3.2.2(b) · May/June 2024 · Paper 1 · National (DBE) · 4 marks
3.2 Given an arithmetic sequence with $T_{1} = 8$ and $T_{2} = 11$ .
A new arithmetic sequence $P$ is formed using the term position and the term value of the given arithmetic sequence. For the new sequence, $P_{8} = 1$ , $P_{11} = 2$ and so forth.
3.2.2(b) Calculate the value of the first term of the new arithmetic sequence.(4)
Show the memo (4 ticks)Memo
1. ✓
  Set up first equation using given sequence data $a + 7 d = 1$
  1 mark · Find a, d or r from two facts
2. ✓
  Set up second equation using given sequence data $a + 10 d = 2$
  1 mark · Find a, d or r from two facts
3. ✓
  Solve for common difference (d) $3 d = 1 ⟹ d = \frac{1}{3}$
  1 mark · Solve simultaneous equations by elimination
4. ✓
  Solve for first term (a) $a + 7 (\frac{1}{3}) = 1 ⟹ a = - \frac{4}{3}$
  1 mark · Find a, d or r from two facts
4. Question 3.1 · March 2022 · Provincial · 5 marks
3.1 The $4^{t h}$ term of an arithmetic sequence is $- 3$ and the $2 0^{t h}$ term is $- 35$ . Determine the common difference and the first term.(5)
Show the memo (5 ticks)Memo
1. ✓
  Set up equations $a + 3 d = - 3$ ... (1)
  $a + 19 d = - 35$ ... (2)
  2 marks · Find a, d or r from two facts
2. ✓
  Equating $a = - 3 - 3 d$ and $a = - 35 - 19 d$
  $- 3 - 3 d = - 35 - 19 d$
  1 mark · Solve simultaneous equations by elimination
3. ✓
  d value $16 d = - 32$
  $d = - 2$
  1 mark · Find a, d or r from two facts
4. ✓
  a value $a = - 3 - 3 (- 2) = 3$
  1 mark · Find a, d or r from two facts
5. Question 1.4 · March 2021 · Provincial · 5 marks
1.4 Atlehang generates a sequence, which he claims that the sequence can be arithmetic and also be geometric if and only if the first term in the sequence is 1. Is he correct? Motivate your answer by showing all your calculations.(5)
Show the memo (5 ticks)Memo
1. ✓
  Set up equations for arithmetic and geometric sequencesArithmetic: $1; 1 + d; 1 + 2 d; ...$
  Geometric: $1; r; r^{2}; ...$
  $r = 1 + d$ ... (1)
  $r^{2} = 1 + 2 d$ ... (2)
  1 mark · Find a, d or r from two facts
2. ✓
  Substitute and solveSubstitute (1) into (2):
  $(1 + d)^{2} = 1 + 2 d$
  $1 + 2 d + d^{2} = 1 + 2 d$
  $d^{2} = 0$
  $d = 0$
  2 marks · Solve linear and non-linear simultaneous equations
3. ✓
  Determine r and concludeIf $d = 0$ , then $r = 1$ .
  Arithmetic sequence: $1; 1; 1; ...$
  Geometric sequence: $1; 1; 1; ...$
  He is correct.
  2 marks · Find a, d or r from two facts
6. Practice · Siyavula Grade 12 Maths · Ex 1.8 · 2 marks
The common difference of an arithmetic series is $3$ .
Explain why there are two possible answers.(2)
Show the memo (2 ticks)Memo
1. ✓
  Series for $n = 13$ $n = 13$ : $a = 96 - 39 = 57$ : $57 + 60 + \dots + 93 = 975$
  1 mark · Find a, d or r from two facts
2. ✓
  Series for $n = 50$ $n = 50$ : $a = - 54$ : $- 54 - 51 - \dots + 93 = 975$
  Two different series fit both conditions.
  1 mark · Find a, d or r from two facts
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue

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