To find the sum of the indicated number of terms of each of the given arithmetic progressions (A.P.), we will use the formula for the sum of the first n terms of an A.P.:
\[
S_n = \frac{n}{2} \left(2a + (n - 1)d\right)
\]
where:
- \(S_n\) is the sum of the first n terms,
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the number of terms.
Let's solve each part step by step.
### (i) A.P.: 5, 2, -1, -4, -7, …, n terms
1. **Identify the first term \(a\)**:
- \(a = 5\)
2. **Calculate the common difference \(d\)**:
- \(d = 2 - 5 = -3\)
3. **Use the sum formula**:
\[
S_n = \frac{n}{2} \left(2 \cdot 5 + (n - 1)(-3)\right)
\]
\[
S_n = \frac{n}{2} \left(10 - 3(n - 1)\right)
\]
\[
S_n = \frac{n}{2} \left(10 - 3n + 3\right)
\]
\[
S_n = \frac{n}{2} \left(13 - 3n\right)
\]
### (ii) A.P.: -1, 1/4, 3/2, …, 81 terms
1. **Identify the first term \(a\)**:
- \(a = -1\)
2. **Calculate the common difference \(d\)**:
- \(d = \frac{1}{4} - (-1) = \frac{1}{4} + 1 = \frac{5}{4}\)
3. **Use the sum formula**:
\[
S_{81} = \frac{81}{2} \left(2 \cdot (-1) + (81 - 1) \cdot \frac{5}{4}\right)
\]
\[
S_{81} = \frac{81}{2} \left(-2 + 80 \cdot \frac{5}{4}\right)
\]
\[
S_{81} = \frac{81}{2} \left(-2 + 100\right)
\]
\[
S_{81} = \frac{81}{2} \cdot 98 = 81 \cdot 49 = 3969
\]
### (iii) A.P.: 2, 4, 6, …, 100 terms
1. **Identify the first term \(a\)**:
- \(a = 2\)
2. **Calculate the common difference \(d\)**:
- \(d = 4 - 2 = 2\)
3. **Use the sum formula**:
\[
S_{100} = \frac{100}{2} \left(2 \cdot 2 + (100 - 1) \cdot 2\right)
\]
\[
S_{100} = 50 \left(4 + 198\right)
\]
\[
S_{100} = 50 \cdot 202 = 10100
\]
### (iv) A.P.: -0.5, -1.0, -1.5, …, 10 terms and 50 terms
1. **Identify the first term \(a\)**:
- \(a = -0.5\)
2. **Calculate the common difference \(d\)**:
- \(d = -1.0 - (-0.5) = -0.5\)
3. **Sum for 10 terms**:
\[
S_{10} = \frac{10}{2} \left(2 \cdot (-0.5) + (10 - 1)(-0.5)\right)
\]
\[
S_{10} = 5 \left(-1 + (-4.5)\right) = 5 \cdot (-5.5) = -27.5
\]
4. **Sum for 50 terms**:
\[
S_{50} = \frac{50}{2} \left(2 \cdot (-0.5) + (50 - 1)(-0.5)\right)
\]
\[
S_{50} = 25 \left(-1 + (-24.5)\right) = 25 \cdot (-25.5) = -637.5
\]
### (v) A.P.: \(x+y, x-y, x-3y, \ldots\), 22 terms
1. **Identify the first term \(a\)**:
- \(a = x + y\)
2. **Calculate the common difference \(d\)**:
- \(d = (x - y) - (x + y) = -2y\)
3. **Use the sum formula**:
\[
S_{22} = \frac{22}{2} \left(2(x + y) + (22 - 1)(-2y)\right)
\]
\[
S_{22} = 11 \left(2x + 2y + 21(-2y)\right)
\]
\[
S_{22} = 11 \left(2x + 2y - 42y\right) = 11 \left(2x - 40y\right) = 22x - 440y
\]
### Summary of Results:
- (i) \(S_n = \frac{n}{2}(13 - 3n)\)
- (ii) \(S_{81} = 3969\)
- (iii) \(S_{100} = 10100\)
- (iv) \(S_{10} = -27.5\), \(S_{50} = -637.5\)
- (v) \(S_{22} = 22x - 440y\)
