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

1 answer

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 \( -487 \).