Question

The 43rd term of an AP is 26,find the first term of progression given that it's term of progression given that it's common difference 13 1/2(half) and also find the 5th term

Answer 1

In an arithmetic progression (AP), the \( n \)-th term can be calculated using the formula:

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

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

Given:
- The 43rd term \( a_{43} = 26 \)
- The common difference \( d = 13.5 \)

We can substitute these values into the formula for the 43rd term:

\[
a_{43} = a + (43 - 1) \cdot d
\]
\[
26 = a + 42 \cdot 13.5
\]

Calculating \( 42 \cdot 13.5 \):

\[
42 \cdot 13.5 = 567
\]

Now substituting this back into the equation:

\[
26 = a + 567
\]

To find \( a \):

\[
a = 26 - 567
\]
\[
a = 26 - 567 = -541
\]

Now we have found the first term \( a = -541 \).

Next, we want to find the 5th term \( a_5 \):

\[
a_5 = a + (5 - 1) \cdot d
\]
\[
a_5 = -541 + 4 \cdot 13.5
\]

Calculating \( 4 \cdot 13.5 \):

\[
4 \cdot 13.5 = 54
\]

Now substituting this back into the equation:

\[
a_5 = -541 + 54
\]
\[
a_5 = -541 + 54 = -487
\]

Thus, the first term of the progression is \( -541 \) and the 5th term is \( -487 \).