The sim of the first 10th terms of an AP is 130. If the fifth term is 3 times the first term. Find

Question

Common difference
First term
Number of term of AP if the last term is 28

Answer 1

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

We are given that the sum of the first 10 terms of the AP is 130:
S10 = 130

We are also given that the fifth term is 3 times the first term:
a + 4d = 3a

Now, let's solve the equation a + 4d = 3a to find the common difference 'd':
4d = 2a
d = 0.5a

Substituting the value of 'd' in the sum formula, we have:
S10 = (10/2)(2a + 9d)
130 = 5(2a + 9(0.5a))
130 = 5(2a + 4.5a)
130 = 5(6.5a)
130 = 32.5a

Solving for 'a', we get:
a = 130/32.5
a = 4

Therefore, the first term is 4.
Since d = 0.5a, the common difference is 0.5 * 4 = 2.

Now, to find the number of terms if the last term is 28, we can use the formula for the nth term of an arithmetic progression:
an = a + (n-1)d

Substituting the known values, we have:
28 = 4 + (n-1)(2)
28 - 4 = 2n - 2
24 = 2n - 2
26 = 2n
n = 13

Therefore, the number of terms in the AP is 13.