The sum of the 5th term of the 7th term of an ap is 52 and the 10th term is 46 find the AP

The AP is, 1, 6, 11, 16, 21, 26, 31, ...

Step-by-step explanation:

From the question;

Sum of fifth term and 7th term is 52 That is; 5th term + 7th term = 52 10th term is 46

We are required to find the AP;

We need to know that in an AP;

nth term = a + (n-1) d, where a is the first term and d is the common difference

Therefore, In this case;

5th term = a + 4d

7th term = a + 6d

Therefore;

5th term + 7th term = 2a + 10d = 52 ... eqn 1

10th term = a + 9d

Thus, a + 9d = 46 ... eqn 2 We can solve eqn 1 and eqn 2 simultaneously to get the value of a and d

2a + 10d = 52

a + 9d = 46

Multiplying the second equation by 2, we get;

2a + 10d = 52

2a + 18d = 92

Eliminating a by subtracting the two equations; we get;

-8d = - 40

d = 5

Solving for a

a + 9d = 46

a = 46 - 9d

= 46 - 9 (5)

= 46 - 45

a = 1

Thus, the first term, a = 1 and the common difference, d = 5

Therefore;

First term = a = 1

Second term = a + d = 1 + 5 = 6

Third term = a + 2d = 1 + 2 (5) = 11

Fourth term = a + 3d = 1 + 3 (5) = 16

Fifth term = a + 4d = 1 + 4 (5) = 21

Sixth term = 1 + 5d = 1 + 5 (5) = 26

Seventh term = 1 + 6d = 1 + 6 (5) = 31

Thus,

The AP is, 1, 6, 11, 16, 21, 26, 31, ...