What is a Mixed Sequence?#
Sometimes a sequence doesn’t behave like a single arithmetic or geometric pattern. The differences aren’t constant, and neither are the ratios. Before you panic — check if the sequence is actually two separate sequences interleaved (woven together, alternating).
How to Spot One#
If a sequence has no constant first difference and no constant ratio, separate the odd-positioned and even-positioned terms and check each group independently.
Take this sequence: $5;\;3;\;10;\;6;\;20;\;12;\;40;\;24;\;\dots$
At first glance, this looks chaotic. But watch what happens when we split it:
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Term | $5$ | $3$ | $10$ | $6$ | $20$ | $12$ | $40$ | $24$ |
Odd positions ($T_1, T_3, T_5, T_7$): $5;\;10;\;20;\;40$
$$\frac{10}{5} = 2, \quad \frac{20}{10} = 2, \quad \frac{40}{20} = 2$$This is geometric with $a = 5$ and $r = 2$.
Even positions ($T_2, T_4, T_6, T_8$): $3;\;6;\;12;\;24$
$$\frac{6}{3} = 2, \quad \frac{12}{6} = 2, \quad \frac{24}{12} = 2$$This is also geometric with $a = 3$ and $r = 2$.
Two perfectly clear patterns hiding inside one “messy” sequence.
1. The Position Mapping — Which Sub-Sequence Does a Term Belong To?#
This is where most students lose marks. You must correctly map between the overall position $n$ and the position within the sub-sequence $k$.
The Rules#
| Overall position $n$ | Type | Position in sub-sequence ($k$) |
|---|---|---|
| Odd ($n = 1, 3, 5, 7, \dots$) | Belongs to the odd sub-sequence | $k = \dfrac{n + 1}{2}$ |
| Even ($n = 2, 4, 6, 8, \dots$) | Belongs to the even sub-sequence | $k = \dfrac{n}{2}$ |
See It With Numbers#
Using our sequence $5;\;3;\;10;\;6;\;20;\;12;\;40;\;24;\;\dots$:
| Overall position $n$ | Odd/Even? | Sub-sequence position $k$ | Sub-sequence term |
|---|---|---|---|
| $n = 1$ | Odd | $k = \frac{1+1}{2} = 1$ | 1st odd term $= 5$ |
| $n = 2$ | Even | $k = \frac{2}{2} = 1$ | 1st even term $= 3$ |
| $n = 3$ | Odd | $k = \frac{3+1}{2} = 2$ | 2nd odd term $= 10$ |
| $n = 4$ | Even | $k = \frac{4}{2} = 2$ | 2nd even term $= 6$ |
| $n = 7$ | Odd | $k = \frac{7+1}{2} = 4$ | 4th odd term $= 40$ |
| $n = 50$ | Even | $k = \frac{50}{2} = 25$ | 25th even term |
| $n = 51$ | Odd | $k = \frac{51+1}{2} = 26$ | 26th odd term |
2. The Full Strategy#
- Check if the sequence is arithmetic or geometric first (it might not be mixed)
- Separate the odd-positioned and even-positioned terms
- Identify each sub-sequence type (arithmetic, geometric, or quadratic)
- Find the general term for each sub-sequence
- Map the overall position $n$ to the correct sub-sequence position $k$
- Calculate the required term
Worked Example 1 — Finding a Specific Term#
Given the sequence $5;\;3;\;10;\;6;\;20;\;12;\;40;\;24;\;\dots$, find $T_{20}$ and $T_{21}$.
Step 1 — Identify sub-sequences:
Odd positions ($T_1, T_3, T_5, T_7, \dots$): $5;\;10;\;20;\;40;\;\dots$ → geometric, $a_{\text{odd}} = 5$, $r_{\text{odd}} = 2$
Even positions ($T_2, T_4, T_6, T_8, \dots$): $3;\;6;\;12;\;24;\;\dots$ → geometric, $a_{\text{even}} = 3$, $r_{\text{even}} = 2$
Step 2 — Find $T_{20}$:
$n = 20$ is even, so $k = \frac{20}{2} = 10$ (the 10th term of the even sub-sequence).
$$T_{20} = a_{\text{even}} \cdot r_{\text{even}}^{k-1} = 3 \cdot 2^{10-1} = 3 \cdot 2^9 = 3 \times 512 = \boxed{1\,536}$$Step 3 — Find $T_{21}$:
$n = 21$ is odd, so $k = \frac{21+1}{2} = 11$ (the 11th term of the odd sub-sequence).
$$T_{21} = a_{\text{odd}} \cdot r_{\text{odd}}^{k-1} = 5 \cdot 2^{11-1} = 5 \cdot 2^{10} = 5 \times 1\,024 = \boxed{5\,120}$$Worked Example 2 — Arithmetic + Geometric Mix#
The sequence $4;\;1;\;7;\;3;\;10;\;9;\;13;\;27;\;\dots$ is a mixed sequence. Find $T_{15}$ and $T_{16}$.
Step 1 — Separate:
Odd positions ($T_1, T_3, T_5, T_7, \dots$): $4;\;7;\;10;\;13;\;\dots$
$$7 - 4 = 3, \quad 10 - 7 = 3, \quad 13 - 10 = 3$$Arithmetic with $a_{\text{odd}} = 4$, $d = 3$.
Even positions ($T_2, T_4, T_6, T_8, \dots$): $1;\;3;\;9;\;27;\;\dots$
$$\frac{3}{1} = 3, \quad \frac{9}{3} = 3, \quad \frac{27}{9} = 3$$Geometric with $a_{\text{even}} = 1$, $r = 3$.
Step 2 — Find $T_{15}$:
$n = 15$ is odd, so $k = \frac{15+1}{2} = 8$.
$$T_{15} = a_{\text{odd}} + (k-1)d = 4 + (8-1)(3) = 4 + 21 = \boxed{25}$$Step 3 — Find $T_{16}$:
$n = 16$ is even, so $k = \frac{16}{2} = 8$.
$$T_{16} = a_{\text{even}} \cdot r^{k-1} = 1 \cdot 3^{8-1} = 3^7 = \boxed{2\,187}$$Worked Example 3 — Finding Which Term Equals a Value#
In the mixed sequence from Example 2, which term equals $31$?
The odd sub-sequence is arithmetic: $T_k^{\text{odd}} = 4 + (k-1)(3) = 3k + 1$
$$3k + 1 = 31 \quad \Rightarrow \quad 3k = 30 \quad \Rightarrow \quad k = 10$$The 10th term of the odd sub-sequence. Overall position:
$$n = 2k - 1 = 2(10) - 1 = 19$$$$\boxed{T_{19} = 31}$$Check: $T_{19}$ is the 10th odd-positioned term. $T_{10}^{\text{odd}} = 4 + 9(3) = 31\;\checkmark$
Now check the even sub-sequence: $T_k^{\text{even}} = 3^{k-1}$
$$3^{k-1} = 31$$Since $3^3 = 27$ and $3^4 = 81$, there is no integer $k$ that works. So $31$ only appears once, at $T_{19}$.
3. Sums of Mixed Sequences#
To find the sum of a mixed sequence, sum each sub-sequence separately.
Worked Example 4 — Sum of the First 20 Terms#
Find the sum of the first 20 terms of $5;\;3;\;10;\;6;\;20;\;12;\;40;\;24;\;\dots$
The first 20 terms contain 10 odd-positioned terms and 10 even-positioned terms.
Sum of odd sub-sequence (geometric, $a = 5$, $r = 2$, $n = 10$):
$$S_{10}^{\text{odd}} = \frac{5(2^{10} - 1)}{2 - 1} = 5(1\,024 - 1) = 5(1\,023) = 5\,115$$Sum of even sub-sequence (geometric, $a = 3$, $r = 2$, $n = 10$):
$$S_{10}^{\text{even}} = \frac{3(2^{10} - 1)}{2 - 1} = 3(1\,023) = 3\,069$$Total:
$$S_{20} = S_{10}^{\text{odd}} + S_{10}^{\text{even}} = 5\,115 + 3\,069 = \boxed{8\,184}$$Worked Example 5 — Sum of Odd Number of Terms#
Find the sum of the first 15 terms of $4;\;1;\;7;\;3;\;10;\;9;\;13;\;27;\;\dots$
15 terms → 8 odd-positioned terms + 7 even-positioned terms (odd positions get one extra since we start with an odd position).
Sum of odd sub-sequence (arithmetic, $a = 4$, $d = 3$, $n = 8$):
$$S_8^{\text{odd}} = \frac{8}{2}[2(4) + 7(3)] = 4[8 + 21] = 4(29) = 116$$Sum of even sub-sequence (geometric, $a = 1$, $r = 3$, $n = 7$):
$$S_7^{\text{even}} = \frac{1(3^7 - 1)}{3 - 1} = \frac{2\,187 - 1}{2} = \frac{2\,186}{2} = 1\,093$$Total:
$$S_{15} = 116 + 1\,093 = \boxed{1\,209}$$How to count: For $N$ total terms: odd sub-sequence has $\lceil \frac{N}{2} \rceil$ terms (round up), even sub-sequence has $\lfloor \frac{N}{2} \rfloor$ terms (round down). For $N = 15$: odd gets $8$, even gets $7$.
4. Other Types of “Mixed” Patterns#
Not all mixed sequences alternate between two interleaved sub-sequences. Here are other patterns you might encounter:
Alternating Signs#
$$5;\;-5;\;5;\;-5;\;\dots$$This is actually geometric with $r = -1$. Or:
$$3;\;-6;\;12;\;-24;\;\dots$$Geometric with $r = -2$. The signs alternate because $r$ is negative — this is not a “mixed” sequence, it’s a single geometric sequence.
Arithmetic-Geometric Combinations#
Some sequences multiply an arithmetic pattern by a geometric one:
$$1;\;4;\;12;\;32;\;80;\;\dots$$If you notice the pattern $T_n = n \cdot 2^{n-1}$, this is an arithmetic-geometric sequence. These are rare in CAPS exams but worth recognising.
Sequences with a “Correction” Term#
$$2;\;5;\;10;\;17;\;26;\;\dots$$This looks like it could be mixed, but check the second differences: $3;\;5;\;7;\;9$ → second differences are $2;\;2;\;2$. This is actually a quadratic sequence, not a mixed one. Always check second differences before assuming a sequence is mixed.
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| Not checking if it’s just arithmetic/geometric/quadratic first | You might be overcomplicating a simple sequence | Always test simple patterns before splitting |
| Wrong position mapping | $T_{50}$ is the 25th even term, not the 50th | Use $k = \frac{n}{2}$ (even) or $k = \frac{n+1}{2}$ (odd) |
| Wrong count for sums of odd total | 15 terms: 8 odd + 7 even, not 7 + 8 | Odd sub-sequence always gets the extra term (since $T_1$ is odd) |
| Assuming both sub-sequences are the same type | One could be arithmetic, the other geometric | Analyse each sub-sequence separately |
| Using the overall position in the sub-sequence formula | $T_{20}$ is the 10th even term, so use $k = 10$ in the formula, not $n = 20$ | Always convert to sub-sequence position first |
💡 Pro Tips for Exams#
1. The “Chaos Test”#
If a sequence seems to follow no pattern, immediately separate odd and even positions. This takes 30 seconds and usually reveals the structure.
Why? Examiners specifically design mixed sequences to look confusing. The difficulty is in recognising that it’s mixed — once you split it, each piece is usually straightforward arithmetic or geometric.
2. Quick Position Check#
Before calculating, always verify your mapping with a known term. For example, if you’ve identified the odd sub-sequence and you want $T_7$, check: $k = \frac{7+1}{2} = 4$, so you need the 4th term of the odd sub-sequence. Count in the original sequence: $T_1, T_3, T_5, T_7$ — yes, that’s the 4th odd-positioned term. $\checkmark$
3. Sum Counting Rule#
For $N$ total terms:
- Odd sub-sequence contributes $\lceil N/2 \rceil$ terms (ceiling — round up)
- Even sub-sequence contributes $\lfloor N/2 \rfloor$ terms (floor — round down)
| $N$ | Odd terms | Even terms |
|---|---|---|
| $10$ | $5$ | $5$ |
| $11$ | $6$ | $5$ |
| $20$ | $10$ | $10$ |
| $21$ | $11$ | $10$ |