Find the indicated term(s) in each of the followng A.P.'s
(i) `-1,-2,-3,-4………….,t_(100)`
(ii) `n-1,n-2,n-3,………….,a_(m)`
(iii) `a=3,d=2,a_(10),a_(n)`

Answer

To solve the problem of finding the indicated terms in each of the given arithmetic progressions (A.P.s), we will use the formula for the nth term of an A.P., which is given by:

\[ a_n = a + (n - 1)d \]

where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

Now, let's solve each part step by step.

### (i) A.P.: -1, -2, -3, -4, ..., \( t_{100} \)

1. **Identify the first term and common difference**:
- First term \( a = -1 \)
- Second term \( a_2 = -2 \)
- Common difference \( d = a_2 - a_1 = -2 - (-1) = -1 \)

2. **Use the formula for the 100th term**:
\[ t_{100} = a + (100 - 1)d \]
\[ t_{100} = -1 + (99)(-1) \]
\[ t_{100} = -1 - 99 = -100 \]

### (ii) A.P.: \( n-1, n-2, n-3, ..., a_m \)

1. **Identify the first term and common difference**:
- First term \( a = n - 1 \)
- Second term \( a_2 = n - 2 \)
- Common difference \( d = a_2 - a_1 = (n - 2) - (n - 1) = -1 \)

2. **Use the formula for the mth term**:
\[ a_m = a + (m - 1)d \]
\[ a_m = (n - 1) + (m - 1)(-1) \]
\[ a_m = n - 1 - (m - 1) = n - m \]

### (iii) A.P.: \( a = 3, d = 2, a_{10}, a_n \)

1. **Find the 10th term**:
\[ a_{10} = a + (10 - 1)d \]
\[ a_{10} = 3 + (9)(2) \]
\[ a_{10} = 3 + 18 = 21 \]

2. **Find the nth term**:
\[ a_n = a + (n - 1)d \]
\[ a_n = 3 + (n - 1)(2) \]
\[ a_n = 3 + 2n - 2 = 2n + 1 \]

### Summary of Results:
- (i) \( t_{100} = -100 \)
- (ii) \( a_m = n - m \)
- (iii) \( a_{10} = 21 \) and \( a_n = 2n + 1 \)