Sum of4th and the 6th terms of an a,p is 42 . While the sum of the 3rd and 9th terms of the same a.p is 52. Find the (a) first term (b)common difference (c) sum of the first ten terms of the progression. Solution

I assume that "a,p" stands for arithmetic progression. Let the nth term be an.

Since a4 + a6 = 2 a5, a5 = 21.
Since a3 + a9 = 2 a6, a6 = 26

The common difference is
a(n+1) - an = a6 - a5 = 5.

The first term is 1, and the progression is
a1 = 1, a2 = 6, a3 = 11, a4 = 16,
a5 = 21, a6 = 26, a7 = 31, a8 = 36,
a9 = 41, a10 = 46...

The sum of the first 10 terms is
7 + 27 + 47 + 67 + 87 = 235

Which of the following describes the sum in terms of p and q?

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

Let's take the general form of an arithmetic progression:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term and 'd' is the common difference.

Using the given information:
1. Sum of the 4th and 6th terms is 42:
(a + 3d) + (a + 5d) = 42
2a + 8d = 42 ------(1)

2. Sum of the 3rd and 9th terms is 52:
(a + 2d) + (a + 8d) = 52
2a + 10d = 52 ------(2)

Now, we have a system of two equations with two variables, (a and d). We can solve this system to find their values.

To solve the system of equations (1) and (2), we will use the method of substitution or elimination. I will use the elimination method here:

Step 1: Multiply equation (1) by 5 and equation (2) by 4 to make the coefficients of 'a' equal:
10a + 40d = 210 ------(3)
8a + 40d = 208 ------(4)

Step 2: Subtract equation (4) from equation (3) to eliminate 'd':
(10a + 40d) - (8a + 40d) = 210 - 208
2a = 2
a = 1

Now, we have found the value of 'a' which is the first term of the AP. Substituting this value back into either equation (1) or (2) will help us find the value of 'd'. Let's substitute it into equation (1):

2(1) + 8d = 42
2 + 8d = 42
8d = 42 - 2
8d = 40
d = 40/8
d = 5

So, the first term (a) of the AP is 1 and the common difference (d) is 5.

To find the sum of the first ten terms of the AP, we can use the formula:

Sum of the first 'n' terms = (n/2) * (2a + (n-1)d)

Substituting the values:
n = 10 (as we want the sum of the first ten terms)
a = 1 (first term)
d = 5 (common difference)

Sum of the first 10 terms = (10/2) * (2 * 1 + (10 - 1) * 5)
= 5 * (2 + 9 * 5)
= 5 * (2 + 45)
= 5 * 47
= 235

Therefore, the sum of the first ten terms of the AP is 235.

To summarize:
(a) The first term of the AP is 1.
(b) The common difference of the AP is 5.
(c) The sum of the first ten terms of the AP is 235.