Question

The sum of the first three term of an ap is 21 and the sum of the first five term of the sequence is 55 calculate the first term, common difference, seventh term, sum of the first ten term

Answer 1

Let the first term of the arithmetic progression (AP) be \( a \) and the common difference be \( d \).

### Step 1: Set up equations
The sum of the first \( n \) terms of an AP is given by the formula:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
From the problem statement:

1. The sum of the first 3 terms is 21:
\[
S_3 = \frac{3}{2} \times (2a + 2d) = 21
\]
Simplifying this, we get:
\[
3(2a + 2d) = 42
\]
\[
2a + 2d = 14 \quad \text{(Equation 1)}
\]
\[
a + d = 7 \quad \text{(Equation 2)}
\]

2. The sum of the first 5 terms is 55:
\[
S_5 = \frac{5}{2} \times (2a + 4d) = 55
\]
Simplifying this, we get:
\[
5(2a + 4d) = 110
\]
\[
2a + 4d = 22 \quad \text{(Equation 3)}
\]

### Step 2: Solve the equations
Now we have two equations:
1. \( 2a + 2d = 14 \) (Equation 1)
2. \( 2a + 4d = 22 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[
(2a + 4d) - (2a + 2d) = 22 - 14
\]
This simplifies to:
\[
2d = 8
\]
Thus, we find:
\[
d = 4
\]

Now substitute \( d = 4 \) into Equation 2 to find \( a \):
\[
a + 4 = 7
\]
\[
a = 3
\]

### Step 3: Find the seventh term
The \( n \)-th term of an AP is given by:
\[
a_n = a + (n-1)d
\]
For the seventh term:
\[
a_7 = a + 6d = 3 + 6 \times 4 = 3 + 24 = 27
\]

### Step 4: Calculate the sum of the first ten terms
Using the sum formula:
\[
S_{10} = \frac{10}{2} \times (2a + 9d)
\]
Calculating:
\[
S_{10} = 5 \times (2 \times 3 + 9 \times 4) = 5 \times (6 + 36) = 5 \times 42 = 210
\]

### Summary of results
- First term \( a = 3 \)
- Common difference \( d = 4 \)
- Seventh term \( a_7 = 27 \)
- Sum of the first ten terms \( S_{10} = 210 \)