To find the first term of the arithmetic progression (AP), let's denote the first term by [tex]\(a\)[/tex] and the common difference by [tex]\(d\)[/tex].
Given:
1. The [tex]\( (4p)^{\text{th}} \)[/tex] term of the AP is 15 more than the [tex]\( (3p)^{\text{th}} \)[/tex] term.
2. The [tex]\( (p+1)^{\text{th}} \)[/tex] term is 18.
For an AP, the [tex]\( n^{\text{th}} \)[/tex] term is given by:
[tex]\[ a_n = a + (n-1)d \][/tex]
Using this formula, write the expressions for the given terms:
### Step 1: Express the [tex]\( (4p)^{\text{th}} \)[/tex] and [tex]\( (3p)^{\text{th}} \)[/tex] terms
- The [tex]\( (4p)^{\text{th}} \)[/tex] term:
[tex]\[ a + (4p-1)d \][/tex]
- The [tex]\( (3p)^{\text{th}} \)[/tex] term:
[tex]\[ a + (3p-1)d \][/tex]
According to the given condition:
[tex]\[ a + (4p-1)d = 15 + \left(a + (3p-1)d\right) \][/tex]
### Step 2: Simplify the above equation
[tex]\[ a + 4pd - d = 15 + a + 3pd - d \][/tex]
Cancel common terms from both sides:
[tex]\[ 4pd - d = 15 + 3pd - d \][/tex]
Simplify by eliminating [tex]\( -d \)[/tex]:
[tex]\[ 4pd = 15 + 3pd \][/tex]
Isolate the term involving [tex]\(pd\)[/tex]:
[tex]\[ 4pd - 3pd = 15 \][/tex]
[tex]\[ pd = 15 \][/tex]
### Step 3: Use the expression for the [tex]\( (p+1)^{\text{th}} \)[/tex] term
The [tex]\( (p+1)^{\text{th}} \)[/tex] term:
[tex]\[ a + (p+1-1)d = a + pd \][/tex]
We are given:
[tex]\[ a + pd = 18 \][/tex]
### Step 4: Substitute [tex]\( pd = 15 \)[/tex] into the above equation
[tex]\[ a + 15 = 18 \][/tex]
[tex]\[ a = 18 - 15 \][/tex]
[tex]\[ a = 3 \][/tex]
Therefore, the first term [tex]\(a\)[/tex] is [tex]\( \boxed{3} \)[/tex].
So, the correct answer is [tex]\( \text{A. } 3 \)[/tex].
