Answer :
To solve this problem, we need to understand that we are dealing with an arithmetic progression (AP). We are given the sum of 16 terms in an AP, which is 56, and we need to find the first term [tex]\(a\)[/tex] and the common difference [tex]\(d\)[/tex].
In an arithmetic progression, the sum of the first [tex]\(n\)[/tex] terms is given by the formula:
[tex]\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \][/tex]
Given that the sum of the first 16 terms ([tex]\(S_{16}\)[/tex]) is 56, we can set up the equation:
[tex]\[ 56 = \frac{16}{2} \times (2a + 15d) \][/tex]
Simplifying, we have:
[tex]\[ 56 = 8 \times (2a + 15d) \][/tex]
Dividing both sides by 8 to isolate the terms involving [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ 7 = 2a + 15d \][/tex]
This equation shows that we have a relationship between the first term [tex]\(a\)[/tex] and the common difference [tex]\(d\)[/tex]. However, with only one equation and two unknowns (one equation involving both [tex]\(a\)[/tex] and [tex]\(d\)[/tex]), we can't determine unique values for [tex]\(a\)[/tex] and [tex]\(d\)[/tex] without additional information about either the first term or the common difference.
In summary, to find the exact values for the first term and the common difference of the AP, more information is needed—typically, either the value of the first term [tex]\(a\)[/tex] or the common difference [tex]\(d\)[/tex].
In an arithmetic progression, the sum of the first [tex]\(n\)[/tex] terms is given by the formula:
[tex]\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \][/tex]
Given that the sum of the first 16 terms ([tex]\(S_{16}\)[/tex]) is 56, we can set up the equation:
[tex]\[ 56 = \frac{16}{2} \times (2a + 15d) \][/tex]
Simplifying, we have:
[tex]\[ 56 = 8 \times (2a + 15d) \][/tex]
Dividing both sides by 8 to isolate the terms involving [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ 7 = 2a + 15d \][/tex]
This equation shows that we have a relationship between the first term [tex]\(a\)[/tex] and the common difference [tex]\(d\)[/tex]. However, with only one equation and two unknowns (one equation involving both [tex]\(a\)[/tex] and [tex]\(d\)[/tex]), we can't determine unique values for [tex]\(a\)[/tex] and [tex]\(d\)[/tex] without additional information about either the first term or the common difference.
In summary, to find the exact values for the first term and the common difference of the AP, more information is needed—typically, either the value of the first term [tex]\(a\)[/tex] or the common difference [tex]\(d\)[/tex].
