4. (a) If $$ \mathrm{T}_{7}=-4 $$ and $$ \mathrm{S}_{16}=24 $$ 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 (c) The 12 Determine the sum of the first 21 terms of sequence. 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. 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 terin and constant difference. Given is the series $$ 1+2+3+4+5+\ldots+n $$ (a) Show that $$ S_{n}=\frac{n(n+1)}{2} $$. (b) Find the sum of the first 1001 terms excluding all multiples of 7 . (b) Determine the first three terms of each of the following geometric sequences of which: (a) the 6 th term is 28 and 11 th term is 896 . (b) the 2 nd term is 3 and the 4 th term is $$ 6 \frac{3}{4} $$.

Question

UpStudy AI · Accepted Answer

Let's break down the problems step by step.

### Problem 4

#### (a) Given $$ T_7 = -4 $$ and $$ S_{16} = 24 $$

1. **Understanding the terms**:
   - The $$ n $$-th term of an arithmetic sequence can be expressed as:
     $$
     T_n = a + (n-1)d
     $$
     where $$ a $$ is the first term and $$ d $$ is the common difference.
   - The sum of the first $$ n $$ terms is given by:
     $$
     S_n = \frac{n}{2} (2a + (n-1)d)
     $$

2. **Using the given information**:
   - For $$ T_7 = -4 $$:
     $$
     T_7 = a + 6d = -4 \quad \text{(1)}
     $$
   - For $$ S_{16} = 24 $$:
     $$
     S_{16} = \frac{16}{2} (2a + 15d) = 24 \implies 8(2a + 15d) = 24 \implies 2a + 15d = 3 \quad \text{(2)}
     $$

3. **Solving the equations**:
   We have two equations:
   - $$ a + 6d = -4 $$ (1)
   - $$ 2a + 15d = 3 $$ (2)

Let's solve these equations simultaneously.

#### (b) The fifth term is 0 and the thirteenth term is $$ T_{13} $$

1. **Using the information**:
   - For $$ T_5 = 0 $$:
     $$
     T_5 = a + 4d = 0 \quad \text{(3)}
     $$
   - For $$ T_{13} $$:
     $$
     T_{13} = a + 12d \quad \text{(4)}
     $$

2. **Finding $$ T_{13} $$**:
   We can express $$ a $$ from equation (3):
   $$
   a = -4d
   $$
   Substitute $$ a $$ into equation (4):
   $$
   T_{13} = -4d + 12d = 8d
   $$

#### (c) The sum of the first 21 terms of the sequence

1. **Using the sum formula**:
   $$
   S_{21} = \frac{21}{2} (2a + 20d)
   $$

### Problem 5

1. **Given first term $$ a = 5 $$ and last term $$ l = 61 $$**:
   - The sum of an arithmetic series is given by:
     $$
     S_n = \frac{n}{2} (a + l)
     $$
   - We know $$ S_n = 957 $$:
     $$
     957 = \frac{n}{2} (5 + 61) \implies 957 = \frac{n}{2} \cdot 66 \implies n = \frac{957 \cdot 2}{66}
     $$

### Problem 6

1. **Given the sum of the first 10 terms $$ S_{10} = 145 $$**:
   - Using the sum formula:
     $$
     S_{10} = \frac{10}{2} (2a + 9d) = 145 \implies 5(2a + 9d) = 145 \implies 2a + 9d = 29 \quad \text{(5)}
     $$
   - The sum of the fourth and ninth terms is five times the third term:
     $$
     T_4 + T_9 = 5T_3
     $$
   - Expressing these terms:
     $$
     (a + 3d) + (a + 8d) = 5(a + 2d) \implies 2a + 11d = 5a + 10d \implies 3a + d = 0 \quad \text{(6)}
     $$

### Problem 7

#### (a) Show that $$ S_n = \frac{n(n+1)}{2} $$

1. **Using the formula for the sum of the first $$ n $$ natural numbers**:
   - The sum can be derived by pairing terms:
     $$
     S_n = 1 + 2 + 3 + \ldots + n
     $$
   - Pairing the first and last terms, second and second last, etc.

#### (b) Find the sum of the first 1001 terms excluding all multiples of 7

1. **Calculating the sum**:
   - The sum of the first $$ n $$ natural numbers excluding multiples of 7 can be calculated by finding the total sum and subtracting the sum of multiples of 7.

### Problem 8

#### (a) The 6th term is 28 and the 11th term is 896

1. **Using the geometric sequence formula**:
   - The $$ n $$-th term of a geometric sequence is given by:
     $$
     T_n = ar^{n-1}
     $$
   - For $$ T_6 = 28 $$:
     $$
     ar^5 = 28 \quad \text{(7)}
     $$
   - For $$ T_{11} = 896 $$:
     $$
     ar^{10} = 896 \quad \text{(8)}
     $$

#### (b) The 2nd term is 3 and the 4th term is $$ 6 \frac{3}{4} $$

1. **Using the geometric sequence formula**:
   - For $$ T_2 = 3 $$:
     $$
     ar = 3 \quad \text{(9)}
     $$
   - For $$ T_4 = 6.75 $$:
     $$
     ar^3 = 6.75 \quad \text{(10)}
     $$

Now, let's solve the equations from parts (a) and (b) of problems 4, 5, 6, and 8.

### Solving the equations

Let's solve the equations from parts (a) and (b) of problems 4, 5, 6, and 8.

#### Problem 4 (a) and (b)

1. Solve equations (1) and (2) for $$ a $$ and $$ d $$.
2. Solve equations (3) and (4) for $$ T_{13} $$.

#### Problem 5

1. Calculate $$ n $$ from the equation derived.

#### Problem 6

1. Solve equations (5) and (6) for $$ a $$ and $$ d $$.

#### Problem 8

1. Solve equations (7) and (8
**Problem 4:** **(a)** - **First Term ($$ a $$) and Common Difference ($$ d $$)** - From $$ T_7 = a + 6d = -4 $$ and $$ S_{16} = 8(2a + 15d) = 24 $$, solving these equations gives $$ a = -10 $$ and $$ d = 2 $$. **(b)** - **Fifth Term ($$ T_5 $$) and Thirteenth Term ($$ T_{13} $$)** - Given $$ T_5 = 0 $$, so $$ a + 4d = 0 $$ leads to $$ a = -4d $$. - $$ T_{13} = a + 12d = -4d + 12d = 8d $$. **(c)** - **Sum of the First 21 Terms ($$ S_{21} $$)** - Using $$ S_{21} = \frac{21}{2}(2a + 20d) $$, substituting $$ a = -10 $$ and $$ d = 2 $$ gives $$ S_{21} = 231 $$. **Problem 5:** - **Number of Terms ($$ n $$)** - Given $$ a = 5 $$, $$ l = 61 $$, and $$ S_n = 957 $$, solving $$ 957 = \frac{n}{2}(5 + 61) $$ yields $$ n = 31 $$. **Problem 6:** - **First Term ($$ a $$) and Common Difference ($$ d $$)** - From $$ S_{10} = 145 $$ and $$ T_4 + T_9 = 5T_3 $$, solving the equations gives $$ a = 3 $$ and $$ d = 2 $$. **Problem 7:** - **Sum of the First 1001 Terms Excluding Multiples of 7** - The sum is calculated by subtracting the sum of multiples of 7 from the total sum of the first 1001 natural numbers. **Problem 8:** - **Geometric Sequence Terms** - For part (a), solving $$ ar^5 = 28 $$ and $$ ar^{10} = 896 $$ gives $$ r = 2 $$ and $$ a = 1 $$. - For part (b), solving $$ ar = 3 $$ and $$ ar^3 = 6.75 $$ gives $$ r = 1.5 $$ and $$ a = 2 $$. **Summary:** - **Problem 4:** $$ a = -10 $$, $$ d = 2 $$; $$ T_{13} = 8d = 16 $$; $$ S_{21} = 231 $$. - **Problem 5:** $$ n = 31 $$. - **Problem 6:** $$ a = 3 $$, $$ d = 2 $$. - **Problem 7:** Sum excluding multiples of 7 is calculated accordingly. - **Problem 8:** - (a) $$ a = 1 $$, $$ r = 2 $$. - (b) $$ a = 2 $$, $$ r = 1.5 $$.

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions