Question

The 5th term of an arithmetic progression is 20 and the 12th term is 41. Find:a) The common difference.b) The value of the first term.c) The sum of the first 8 terms

Answer 1

Given:

The 5th term of an arithmetic progression is 20.

The 12th term of an arithmetic progression 41.

`\begin{gathered} a_5=20 \\ a+(5-1)d=20 \\ a+4d=20\ldots\ldots\ldots(1) \\ a_(12)=41 \\ a+(12-1)d=41 \\ a+11d=41\ldots\ldots\ldots(2) \end{gathered}`

a) To find the common difference:

Subtract equation (1) from (2), we get

`\begin{gathered} 7d=21 \\ d=3 \end{gathered}`

Hence, the common difference is 3.

b) To find the first term:

Substitute d=3 in equation (1), we get

`\begin{gathered} a+4(3)=20 \\ a=20-12 \\ a=8 \end{gathered}`

Hence, the first term is 8.

c) To find the sum of the first 8 terms:

Using the sum formula,

`\begin{gathered} S_n=(n)/(2)\lbrack2a+(n-1)d\rbrack \\ S_8=(8)/(2)\lbrack2(8)+(8-1)3\rbrack \\ =4\lbrack16+21\rbrack \\ =4(37) \\ =148 \end{gathered}`

Hence, the sum of the first 8 terms is 148.