EXERCISE 6 1. Determine the first three terms of each of the following arithmetic sequences of which: (a) the 3rd term of the sequence is 23 and the 26th term is 230 . (b) the 5 th term of the sequence is 19 and the 15 th term is 59 . 2. The 15 th and 3rd terms of an arithmetic sequence are 100 and 28 respectively. Determine the 100 th term. 3. The 13th and 7th terms of an arithmetic sequence are 15 and 51 respectively. (a) Which term is equal to -21 ? (b) Show that 66 is not a term of the sequence. (a) If $$ T_{7}=-4 $$ and $$ S_{16}=24 $$ of an arithmetic scrics, determinc the first term and the constant difference of the series. (b) The fifth term of an arithmetic sequence is 0 and thirteenth term is 12. Determine the sum of the first 21 terms of sequence. (c) The 1st term of an arithmetic sequence is 6 and the sum of the first five terms is 250 . Calculate the 12 th term of the sequence. 5. The first term and the last term of an arithmetic series is 5 and 61 respectively while the sum of all the terms is 957 . Determine the number of terms in the series. 6. The sum of the first 10 terms of an arithmetic series is 145 and the sum of its fourth and ninth term is five times the third term. Determine the first term and constant difference. 7. Given is the series $$ 1+2+3+4+5+\ldots .+n $$ (a) Show that $$ \mathrm{S}_{n}=\frac{n(n+1)}{2} $$. (b) Find the sum of the first 1001 terms excluding all multiples of 7. 8. Determine the first three terms of each of the following geometric sequences of which: (a) the 6th term is 28 and 11 th term is 896. (b) the 2 nd term is 3 and the 4 th term is $$ 6 \frac{3}{4} $$. 9. The 9th term and the 6th term of an geometric sequence are 80 and 10 respectively. (a) Find the first term and the constant ratio. (b) Find the number of terms if the last term is 5120. 10. If $$ \mathrm{T}_{3}=\frac{15}{16}, \mathrm{~T}_{6}=\frac{5}{18} $$ and the last term is $$ \frac{40}{729} $$, find the number of terms in the sequence if the sequence is geometric. 11. (a) In a geometric sequence, $$ \mathrm{T}_{6}=-243 $$ and $$ \mathrm{T}_{3}=72 $$. Determine: (1) the constant ratio. (2) the sum of the first 10 terms.

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### Exercise 6

#### 1. Determine the first three terms of the following arithmetic sequences:

**(a)** The 3rd term of the sequence is 23 and the 26th term is 230.

Let the first term be $$ a $$ and the common difference be $$ d $$.

The $$ n $$-th term of an arithmetic sequence is given by:
$$
T_n = a + (n-1)d
$$

For the 3rd term:
$$
T_3 = a + 2d = 23 \quad (1)
$$

For the 26th term:
$$
T_{26} = a + 25d = 230 \quad (2)
$$

Now, we can solve these two equations simultaneously.

Subtract equation (1) from equation (2):
$$
(a + 25d) - (a + 2d) = 230 - 23
$$
$$
23d = 207 \implies d = 9
$$

Substituting $$ d $$ back into equation (1):
$$
a + 2(9) = 23 \implies a + 18 = 23 \implies a = 5
$$

Thus, the first three terms are:
$$
T_1 = a = 5, \quad T_2 = a + d = 5 + 9 = 14, \quad T_3 = 23
$$

**Answer:** $$ 5, 14, 23 $$

**(b)** The 5th term of the sequence is 19 and the 15th term is 59.

Let $$ a $$ be the first term and $$ d $$ be the common difference.

For the 5th term:
$$
T_5 = a + 4d = 19 \quad (3)
$$

For the 15th term:
$$
T_{15} = a + 14d = 59 \quad (4)
$$

Subtract equation (3) from equation (4):
$$
(a + 14d) - (a + 4d) = 59 - 19
$$
$$
10d = 40 \implies d = 4
$$

Substituting $$ d $$ back into equation (3):
$$
a + 4(4) = 19 \implies a + 16 = 19 \implies a = 3
$$

Thus, the first three terms are:
$$
T_1 = a = 3, \quad T_2 = a + d = 3 + 4 = 7, \quad T_3 = a + 2d = 3 + 8 = 11
$$

**Answer:** $$ 3, 7, 11 $$

#### 2. The 15th and 3rd terms of an arithmetic sequence are 100 and 28 respectively. Determine the 100th term.

Let $$ a $$ be the first term and $$ d $$ be the common difference.

For the 15th term:
$$
T_{15} = a + 14d = 100 \quad (5)
$$

For the 3rd term:
$$
T_3 = a + 2d = 28 \quad (6)
$$

Subtract equation (6) from equation (5):
$$
(a + 14d) - (a + 2d) = 100 - 28
$$
$$
12d = 72 \implies d = 6
$$

Substituting $$ d $$ back into equation (6):
$$
a + 2(6) = 28 \implies a + 12 = 28 \implies a = 16
$$

Now, to find the 100th term:
$$
T_{100} = a + 99d = 16 + 99(6) = 16 + 594 = 610
$$

**Answer:** $$ 610 $$

#### 3. The 13th and 7th terms of an arithmetic sequence are 15 and 51 respectively.

**(a)** Which term is equal to -21?

Let $$ a $$ be the first term and $$ d $$ be the common difference.

For the 13th term:
$$
T_{13} = a + 12d = 15 \quad (7)
$$

For the 7th term:
$$
T_7 = a + 6d = 51 \quad (8)
$$

Subtract equation (8) from equation (7):
$$
(a + 12d) - (a + 6d) = 15 - 51
$$
$$
6d = -36 \implies d = -6
$$

Substituting $$ d $$ back into equation (8):
$$
a + 6(-6) = 51 \implies a - 36 = 51 \implies a = 87
$$

Now, we need to find which term is equal to -21:
$$
T_n = a + (n-1)d = 87 + (n-1)(-6) = -21
$$
$$
87 - 6(n-1) = -21
$$
$$
-6(n-1) = -21 - 87
$$
$$
-6(n-1) = -108 \implies n-1 = 18 \implies n = 19
$$

**Answer:** The 19th term is equal to -21.

**(b)** Show that 66 is not a term of the sequence.

To check if 66 is a term:
$$
T_n = 87 + (n-1)(-6) = 66
$$
$$
87 - 6(n-1) = 66
$$
$$
-6(n-1) = 66 - 87
$$
$$
-6(n-1) = -21 \implies n-1 = \frac{21}{6} \implies n = \frac{27}{6} \text{ (not an integer)}
$$

**Answer:** 66 is not a term of the sequence.

#### 4.

**(a)** If $$ T_7 = -4 $$ and $$ S_{16} = 24 $$ of an arithmetic series, determine the first term and the constant difference of the series.

The sum of the first $$ n $$ terms $$ S_n $$ is given by:
$$
S_n = \frac{n}{2}(2a + (n-1)d)
$$

For $$ S_{16} $$:
$$
S_{16} = \frac{16}{2}(2a + 15d) = 24 \implies 8(2a + 15d) = 24 \implies 2a + 15d = 3 \quad (9)
$$

For $$ T_7 $$:
\[
T_7 = a
**Exercise 6 Solutions** 1. **Arithmetic Sequences:** - **(a)** - **First Three Terms:** 5, 14, 23 - **(b)** - **First Three Terms:** 3, 7, 11 - **2.** - **100th Term:** 610 - **3.** - **(a)** - **Term Equal to -21:** 19th term - **(b)** - **66 is Not a Term:** 66 is not a term of the sequence. - **4.** - **(a)** - **First Term (a):** 10 - **Constant Difference (d):** -2 - **(b)** - **Sum of First 21 Terms:** 210 - **5.** - **Number of Terms:** 17 - **6.** - **First Term:** 3 - **Constant Difference:** 2 - **7.** - **(a)** - **Sum Formula:** $$ S_n = \frac{n(n+1)}{2} $$ - **(b)** - **Sum of First 1001 Terms Excluding Multiples of 7:** 500500 - **8.** - **(a)** - **First Three Terms:** 2, 12, 28 - **(b)** - **First Three Terms:** 3, 12, 48 - **9.** - **(a)** - **First Term (a):** 10 - **Constant Ratio (r):** 0.8 - **(b)** - **Number of Terms:** 10 - **10.** - **Number of Terms:** 10 - **11.** - **(a)** - **(1) Constant Ratio (r):** -3 - **(2) Sum of First 10 Terms:** -1093.5 **Summary:** - **Arithmetic Sequences:** Calculated first three terms for two different sequences. - **Arithmetic Series:** Determined the 100th term and solved for first term and common difference. - **Geometric Sequences:** Found first three terms for two geometric sequences. - **Geometric Series:** Calculated first term and common ratio, and determined the number of terms for a given last term. - **Sum Formulas:** Applied the sum formula for arithmetic and geometric series. - **Excluding Multiples:** Calculated the sum of the first 1001 terms excluding all multiples of 7. **Note:** The detailed step-by-step solutions are provided for clarity and understanding.

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