SOLUTION: Work out the next 5 terms in the number sequences below and explain your calculation using 'n' as your first term; e.g.n+1,n+2,n+3,etc: a. 297;290;276;255;..... b. 1;10;100;.

Question 1165192: Work out the next 5 terms in the number sequences below and explain your calculation using 'n' as your first term; e.g.n+1,n+2,n+3,etc:
a. 297;290;276;255;.....
b. 1;10;100;.....
c. 3;9;81;.....
d. 24;25;27;30;.....
Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
Here are the next 5 terms for each number sequence, along with the explanation of the calculation using a term-to-term rule based on its position in the sequence, $n$.
---
## a. 297; 290; 276; 255; **227; 192; 150; 101; 45**
### Analysis:
This sequence is based on a decreasing difference that increases by a fixed amount each time (a quadratic sequence).
| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |
| :---: | :---: | :---: | :---: |
| $T_1$ | 297 | - | - |
| $T_2$ | 290 | -7 | - |
| $T_3$ | 276 | -14 | **-7** |
| $T_4$ | 255 | -21 | **-7** |
| $T_5$ | **227** | -28 | **-7** |
| $T_6$ | **192** | -35 | **-7** |
| $T_7$ | **150** | -42 | **-7** |
| $T_8$ | **101** | -49 | **-7** |
| $T_9$ | **45** | -56 | **-7** |
### Next 5 Terms:
227, 192, 150, 101, 45
### Term-to-Term Rule:
The difference decreases by 7 each time, starting with 7.
* Next term ($n+1$): **$n - 28$**
* $n+2$: **$(n-28) - 35$**
* $n+3$: **$(n-28-35) - 42$**
* ...and so on.
---
## b. 1; 10; 100; **1,000; 10,000; 100,000; 1,000,000; 10,000,000**
### Analysis:
This is a geometric sequence where each term is 10 times the previous term.
### Next 5 Terms:
1,000, 10,000, 100,000, 1,000,000, 10,000,000
### Term-to-Term Rule:
The next term is the previous term multiplied by 10.
* Next term ($n+1$): **$n \times 10$**
* $n+2$: **$(n \times 10) \times 10$**
* $n+3$: **$(n \times 10 \times 10) \times 10$**
* ...and so on.
---
## c. 3; 9; 81; **6,561; 43,046,721; $1.85 \times 10^{15}$; $3.40 \times 10^{30}$; $1.15 \times 10^{61}$**
### Analysis:
This sequence is generated by squaring the previous term.
* $3^2 = 9$
* $9^2 = 81$
* $81^2 = 6,561$
### Next 5 Terms:
1. $81^2 = \mathbf{6,561}$
2. $6,561^2 = \mathbf{43,046,721}$
3. $43,046,721^2 = \mathbf{1,853,020,188,851,841} \quad (\approx 1.85 \times 10^{15})$
4. $(\approx 1.85 \times 10^{15})^2 \approx \mathbf{3.43 \times 10^{30}}$
5. $(\approx 3.43 \times 10^{30})^2 \approx \mathbf{1.17 \times 10^{61}}$
### Term-to-Term Rule:
The next term is the square of the previous term.
* Next term ($n+1$): **$n^2$**
* $n+2$: **$(n^2)^2 = n^4$**
* $n+3$: **$((n^2)^2)^2 = n^8$**
* ...and so on.
---
## d. 24; 25; 27; 30; **34; 39; 45; 52; 60**
### Analysis:
This sequence is based on an increasing difference (an arithmetic progression of differences).
| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |
| :---: | :---: | :---: | :---: |
| $T_1$ | 24 | - | - |
| $T_2$ | 25 | +1 | - |
| $T_3$ | 27 | +2 | **+1** |
| $T_4$ | 30 | +3 | **+1** |
| $T_5$ | **34** | +4 | **+1** |
| $T_6$ | **39** | +5 | **+1** |
| $T_7$ | **45** | +6 | **+1** |
| $T_8$ | **52** | +7 | **+1** |
| $T_9$ | **60** | +8 | **+1** |
### Next 5 Terms:
34, 39, 45, 52, 60
### Term-to-Term Rule:
The difference increases by 1 each time, starting with 1.
* Next term ($n+1$): **$n + 4$**
* $n+2$: **$(n+4) + 5$**
* $n+3$: **$(n+4+5) + 6$**
* ...and so on.
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