Sure, let's find the first four terms of the arithmetic progression given the details:
Given:
- The arithmetic progression (AP) contains 20 terms.
- The first term ([tex]\( a_1 \)[/tex]) is 2.
- The last term ([tex]\( a_{20} \)[/tex]) is 78.
To find the common difference [tex]\( d \)[/tex] of the AP, we use the formula for the nth term of an arithmetic progression:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
For the 20th term [tex]\( a_{20} \)[/tex]:
[tex]\[ a_{20} = a_1 + (20-1)d \][/tex]
Substituting the given values:
[tex]\[ 78 = 2 + 19d \][/tex]
Now, solving for [tex]\( d \)[/tex]:
[tex]\[ 78 - 2 = 19d \][/tex]
[tex]\[ 76 = 19d \][/tex]
[tex]\[ d = \frac{76}{19} \][/tex]
[tex]\[ d = 4 \][/tex]
Now we know the common difference [tex]\( d \)[/tex] is 4. We can find the first four terms of the AP.
1. The first term [tex]\( a_1 \)[/tex] is given as 2.
2. The second term [tex]\( a_2 \)[/tex] is computed as:
[tex]\[ a_2 = a_1 + d \][/tex]
[tex]\[ a_2 = 2 + 4 \][/tex]
[tex]\[ a_2 = 6 \][/tex]
3. The third term [tex]\( a_3 \)[/tex] is:
[tex]\[ a_3 = a_1 + 2d \][/tex]
[tex]\[ a_3 = 2 + 2 \cdot 4 \][/tex]
[tex]\[ a_3 = 2 + 8 \][/tex]
[tex]\[ a_3 = 10 \][/tex]
4. The fourth term [tex]\( a_4 \)[/tex] is:
[tex]\[ a_4 = a_1 + 3d \][/tex]
[tex]\[ a_4 = 2 + 3 \cdot 4 \][/tex]
[tex]\[ a_4 = 2 + 12 \][/tex]
[tex]\[ a_4 = 14 \][/tex]
So, the first four terms of the arithmetic progression are:
[tex]\[ 2, 6, 10, 14 \][/tex]
The common difference is 4, and the first four terms of the AP are [tex]\( 2, 6, 10, \)[/tex] and [tex]\( 14 \)[/tex].
