if the 2nd term and the 4th term of an arithmetic progression are 15 and 23 respectively, find the first term and the common difference

just use the formulas you know:

a+d = 15
a+3d = 23

Note that T2 and T4 differ by 2d=8.

To find the first term and common difference of an arithmetic progression given the second term and fourth term, you need to use the formula for the term of an arithmetic progression.

The formula for the nth term of an arithmetic progression is:

an = a1 + (n-1)d,

where:
an = nth term,
a1 = first term,
n = position of the term in the progression, and
d = common difference.

We are given that the 2nd term (n=2) is 15 and the 4th term (n=4) is 23.

Let's use this information to find the first term (a1) and the common difference (d).

Using the formula, we substitute n=2 and an=15:
15 = a1 + (2-1)d

Simplifying the equation:
15 = a1 + d

Next, using the formula again, we substitute n=4 and an=23:
23 = a1 + (4-1)d

Simplifying the equation:
23 = a1 + 3d

Now, we have two equations:
15 = a1 + d
23 = a1 + 3d

We can solve this system of equations to find the values of a1 and d.

One way to solve this is by subtracting the first equation from the second equation to eliminate a1:
23 - 15 = (a1 + 3d) - (a1 + d)
8 = 2d

Dividing both sides by 2, we get:
4 = d

Now that we have the value of d, we substitute it back into the first equation to find a1:
15 = a1 + 4

Simplifying the equation:
a1 = 15 - 4
a1 = 11

Therefore, the first term (a1) is 11 and the common difference (d) is 4.