The 14th term of an ap is 96 while the 25th term is 173.find the 19th term,2 sum of 13th and 56th term.3 product of 6th and 13th term.

given:

a+13d = 96
a+24d = 173
solve for a and d, and then you want
a_19 = a+18d
a_13 + a_56 = a+12d + a+55d
(a+5d)(a+12d)

To find the 19th term of an arithmetic progression (AP) given the 14th and 25th terms, we need to find the common difference first.

Step 1: Find the common difference (d):
We know that the 14th term (a14) is 96 and the 25th term (a25) is 173.
Using the formula for the nth term of an AP, we can write:
a14 = a1 + (14 - 1) * d
96 = a1 + 13d ...(1)

a25 = a1 + (25 - 1) * d
173 = a1 + 24d ...(2)

Now we have two equations with two unknowns (a1 and d). We can solve these equations simultaneously to find the values of a1 and d.

Step 2: Solve the equations (1) and (2) simultaneously:
Subtracting equation (1) from equation (2), we get:
173 - 96 = (a1 + 24d) - (a1 + 13d)
77 = 11d

Divide both sides of the equation by 11:
d = 77/11
d = 7
Now that we have found the value of the common difference (d), we can proceed to find the 19th term and the sum of the 13th and 56th terms, as well as the product of the 6th and 13th terms.

Step 3: Find the 19th term (a19):
We can use the formula for the nth term of an AP to find the 19th term:
a19 = a1 + (19 - 1) * d
a19 = a1 + 18d

Since we don't have the value of a1, we need to find it using one of the given terms and the common difference. Let's use the 14th term (a14):
a14 = a1 + (14 - 1) * d
96 = a1 + 13d

Substituting the value of d we found earlier, we can solve for a1:
96 = a1 + 13(7)
96 = a1 + 91
a1 = 96 - 91
a1 = 5

Now we can find the 19th term:
a19 = a1 + 18d
a19 = 5 + 18(7)
a19 = 5 + 126
a19 = 131

Therefore, the 19th term of the arithmetic progression is 131.

Step 4: Find the sum of the 13th and 56th terms:
To find the sum of any two terms in an arithmetic progression, we can use the formula:
Sum of terms = (n/2) * [2a1 + (n-1)d]

Let's find the sum of the 13th and 56th terms:
Sum of 13th and 56th terms = (2/2) * [2a1 + (56-13)d]
Sum of 13th and 56th terms = a1 + (43d)

Substituting the values we found earlier:
Sum of 13th and 56th terms = 5 + (43 * 7)
Sum of 13th and 56th terms = 5 + 301
Sum of 13th and 56th terms = 306

Therefore, the sum of the 13th and 56th terms is 306.

Step 5: Find the product of the 6th and 13th terms:
To find the product of two terms in an arithmetic progression, we simply multiply them together.
Let's find the product of the 6th and 13th terms:
Product of 6th and 13th terms = a6 * a13

To find a6, we can use the formula for the nth term of an AP:
a6 = a1 + (6 - 1) * d
a6 = a1 + 5d

Substituting the values we found earlier:
a6 = 5 + 5(7)
a6 = 5 + 35
a6 = 40

To find a13, we can use the formula for the nth term of an AP:
a13 = a1 + (13 - 1) * d
a13 = a1 + 12d

Substituting the values we found earlier:
a13 = 5 + 12(7)
a13 = 5 + 84
a13 = 89

Now we can find the product of the 6th and 13th terms:
Product of 6th and 13th terms = a6 * a13
Product of 6th and 13th terms = 40 * 89
Product of 6th and 13th terms = 3,560

Therefore, the product of the 6th and 13th terms is 3,560.

To find the 19th term of an arithmetic progression (AP), we'll use the formula:

\[ a_n = a + (n-1)d \]
where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 1: Find the first term (\( a \))
Since the 14th term of the AP is given as 96, we can substitute \( n = 14 \) into the formula:
\[ 96 = a + (14-1)d \]
Simplifying, we get:
\[ 96 = a + 13d \quad \text{(eq. 1)} \]

Step 2: Find the common difference (\( d \))
Similarly, we'll substitute the 25th term of the AP as 173 and \( n = 25 \) into the formula:
\[ 173 = a + (25-1)d \]
Simplifying, we get:
\[ 173 = a + 24d \quad \text{(eq. 2)} \]

From equations (1) and (2), we have a system of two linear equations with two variables (a and d). We can solve this system to find the values of a and d.

Solving the system of equations (1) and (2):
Subtracting equation (1) from equation (2) to eliminate 'a', we get:
\[ 173 - 96 = 24d - 13d \]
\[ 77 = 11d \]
\[ d = \frac{77}{11} \]
\[ d = 7 \]

Substituting the value of \( d \) back into equation (1), we can solve for \( a \):
\[ 96 = a + 13(7) \]
\[ 96 = a + 91 \]
\[ a = 96 - 91 \]
\[ a = 5 \]

Now that we have the values of \( a \) and \( d \), we can find the 19th term, 2 times the sum of the 13th and 56th term, and the product of the 6th and 13th term.

To find the 19th term:
\[ a_{19} = 5 + (19-1)(7) \]
\[ a_{19} = 5 + 18(7) \]
\[ a_{19} = 5 + 126 \]
\[ a_{19} = 131 \]

To find 2 times the sum of the 13th and 56th term:
\[ 2(a_{13} + a_{56}) = 2[(5 + (13-1)(7)) + (5 + (56-1)(7))] \]
\[ 2(a_{13} + a_{56}) = 2[5 + 12(7) + 5 + 55(7)] \]
\[ 2(a_{13} + a_{56}) = 2[5 + 84 + 5 + 385] \]
\[ 2(a_{13} + a_{56}) = 2(479) \]
\[ 2(a_{13} + a_{56}) = 958 \]

To find the product of the 6th and 13th term:
\[ (a_6) \cdot (a_{13}) = (5 + (6-1)(7))(5 + (13-1)(7)) \]
\[ (a_6) \cdot (a_{13}) = (5 + 5(7))(5 + 12(7)) \]
\[ (a_6) \cdot (a_{13}) = (5 + 35)(5 + 84) \]
\[ (a_6) \cdot (a_{13}) = (40)(89) \]
\[ (a_6) \cdot (a_{13}) = 3560 \]

Therefore, the 19th term is 131, 2 times the sum of the 13th and 56th term is 958, and the product of the 6th and 13th term is 3560.