the 11th term of an AP is 4 times its second term .the sum of the first 7 terms of the same progression is 175;a,find the first term and the common difference of the progression

I need more explanation

Using the usual formulas, you have

a+10d = 4(a+d)
7/2 (2a+6d) = 175
Now just solve for a and d

Let's solve this step-by-step:

Step 1: Understanding the problem.
We have an arithmetic progression (AP) where the 11th term is 4 times the second term. Additionally, we know that the sum of the first 7 terms of the AP is 175.

Step 2: Setting up the equations.
Let's assume that the first term of the AP is 'a', and the common difference is 'd'.

We know that the 11th term is 4 times the second term, so we can write the equation:
a + 10d = 4(a + d) ... (Equation 1)

To find the sum of the first 7 terms of the AP, we can use the formula:
Sum = (n/2) * (2a + (n - 1)d), where n is the number of terms.
Plugging in the given values, we get:
175 = (7/2) * (2a + 6d) ... (Equation 2)

Step 3: Solving the equations.
Let's simplify Equation 1:
a + 10d = 4a + 4d
3a = 6d
a = 2d ... (Equation 3)

Substitute Equation 3 into Equation 2:
175 = (7/2) * (2(2d) + 6d)
175 = (7/2) * (4d + 6d)
175 = (7/2) * (10d)
175 = 35d
d = 5

Substitute the value of d into Equation 3:
a = 2(5)
a = 10

So, the first term of the AP is 10 and the common difference is 5.

To find the first term and the common difference of the arithmetic progression (AP), we can use the given information.

Let's denote the first term of the AP as 'a' and the common difference as 'd'.

According to the problem, the 11th term of the AP is 4 times its second term. Mathematically, we can write this as:

a + 10d = 4(a + d)

Simplifying this equation will help us find the value of 'a' in terms of 'd'.

Expanding the equation: a + 10d = 4a + 4d
Combining like terms: 10d - 4d = 4a - a
Simplifying further: 6d = 3a

Now, we need to use the second given information. The sum of the first 7 terms of the AP is 175. The sum of an arithmetic progression can be calculated using the formula:

Sum = (n/2) * [2a + (n-1) * d]

Since the sum is given as 175 and we want the sum of the first 7 terms, n = 7 in this case. Plugging in the values:

175 = (7/2) * [2a + (7-1) * d]
175 = 3.5 * [2a + 6d]
Dividing both sides by 3.5:

50 = 2a + 6d

We now have two equations:

6d = 3a -- Equation 1
2a + 6d = 50 -- Equation 2

We can solve these simultaneous equations to find the values of 'a' and 'd'.

From Equation 1, we can express 'a' in terms of 'd':

a = (6d)/3
a = 2d

Substituting this value of 'a' in Equation 2:

2(2d) + 6d = 50
4d + 6d = 50
10d = 50
d = 5

Now that we have the value of 'd', we can find the value of 'a' using Equation 1:

a = 2d
a = 2(5)
a = 10

Therefore, the first term of the AP is 10 and the common difference is 5.