Given an Arithmetic Progression (A.P), the last term is (4), the common difference is (2), and the sum of all the terms in the A.P is (-14). We are required to find the number of terms in this A.P and also determine the sequence itself.

Step 1: Establishing the Relationship Between the Terms

In an A.P., the nth term (denoted as a_n) is given by the formula:

a_n = a + (n-1)d

where:

(a) is the first term,

(d) is the common difference, and

(n) is the number of terms.

We know that the last term (a_n = 4), and the common difference (d = 2). Substituting these into the formula:

4 = a + (n-1) 2

Expanding this, we get:

4 = a + 2n – 2

Adding 2 to both sides,

we obtain:

6 = a + 2n

Solving for (a), the first term:

a = 6 – 2n

This equation (let’s call it Equation 1) gives us the first term (a) in terms of the number of terms (n).

Step 2: Using the Sum of the A.P :

The sum (S_n )of the first (n) terms of an A.P. is given by:

We know (Sn = -14), (a_n = 4), and substituting (a = 6 – 2n) from Equation 1:

-14 = (n/2)[(6 – 2n) + 4]

Simplifying the expression inside the bracket:

-14 = (n/2)(10 – 2n)

Multiplying both sides by 2 to eliminate the fraction:

-28 = n(10 – 2n)

Expanding and rearranging terms gives us a quadratic equation:

2n² – 10n – 28 = 0

Step 3: Solving the Quadratic Equation

Method 1: Quadratic Formula :

Since (n) must be a positive integer, we have (n = 7).

Method 2: Splitting the Middle Term:

We can also solve the quadratic equation by splitting the middle term.

The quadratic equation is:

2n² – 10n – 28 = 0

First, we need to find two numbers that multiply to the product of the coefficient of n²(which is 2) and the constant term (-28) (i.e., (2 X -28 = -56)), and also add up to the middle coefficient (-10).

The two numbers that satisfy these conditions are (-14) and (4):

2n²– 14n + 4n – 28 = 0

Now, we can group the terms:

(2n²– 14n) + (4n – 28) = 0

Factor each group:

2n(n – 7) + 4(n – 7) = 0

Now, factor out the common binomial factor ((n – 7)):

(n – 7)(2n + 4) = 0

This gives us two possible solutions:

n – 7 = 0 or 2n + 4 = 0

Solving each equation:

n = 7 or 2n = -4

n = -2

Since (n) must be a positive integer, we discard (n = -2) and are left with (n = 7).

Step 4: Finding the First Term and the A.P. Sequence

Substituting (n = 7) back into Equation 1 to find the first term (a)

:

a = 6 – 2(7) = 6 – 14 = -8 ]

Now that we have (a = -8), (d = 2), and (n = 7),

we can list the terms of the A.P. as:

a , a+d , a+2d , a+3d, a+4d , a+5d , a+6d

= -8 , -8+2 , -8+2(2) , -8+3(2) , -8+4(2) , -8+5(2) , -8+6(2)

= -8, -6, -4, -2, 0, 2, 4

Conclusion :

Thus, the Arithmetic Progression is (-8, -6, -4, -2, 0, 2, 4), and the number of terms in this sequence is (7).