Sum of first six term of an ap is 46 ratio of its 10th term to that of 30th term is 1:3. Find its 13th term
what is there
To find the 13th term of the arithmetic progression (AP), we need to use the given information about the sum of the first six terms and the ratio between the 10th and 30th terms.
Let's break it down step by step:
Step 1: Finding the common difference (d)
The sum of the first six terms of an AP is given by the formula: S = (n/2)(2a + (n-1)d)
where S is the sum, a is the first term, n is the number of terms, and d is the common difference.
In this case, we are given that the sum of the first six terms is 46. So, we can rewrite the formula as follows:
46 = (6/2)(2a + (6-1)d)
Simplifying further:
46 = 3(2a + 5d)
46 = 6a + 15d
Step 2: Finding the 10th term (a10) and 30th term (a30)
We are also given that the ratio between the 10th and 30th terms is 1:3. So, we can write:
a10/a30 = 1/3
Using the formula for the nth term of an AP, we have:
a10 = a + (10-1)d
a30 = a + (30-1)d
Step 3: Setting up equations and solving them
We can substitute these values into our previous equation to form a system of linear equations:
46 = 6a + 15d (equation 1)
a10/a30 = 1/3 (equation 2)
Substituting the expressions for a10 and a30:
a + 9d = (1/3)(a + 29d) (Multiplying both sides of equation 2 by 3)
Simplifying further:
3a + 27d = a + 29d
2a = 2d
a = d
Step 4: Finding the 13th term (a13)
Since a = d, we can substitute d for a in equation 1:
46 = 6d + 15d
46 = 21d
d = 46/21
Now, we can find the 13th term (a13) using the formula for the nth term:
a13 = a + (13-1)d
a13 = d + (12)d
a13 = 13d
Substituting the value of d:
a13 = 13(46/21)
Simplifying:
a13 ≈ 28
Therefore, the 13th term of the arithmetic progression is approximately 28.
just use what you know about APs
6/2 (2a+5d) = 46
(a+9d)/(a+29d) = 1/3
solve for a and d, then find a+12d