By using the formula for the sum of an A.P., identifying the middle term position, expressing the last term in terms of the first term and common difference, and utilizing the known middle term value, we found that the sum of the 13 terms in the A.P. is 169. This solution demonstrates how to solve similar problems involving A.P.s with specific information about the middle term.
Find the sum (S) of 13 terms in an arithmetic progression (A.P.) where the middle term (a6) is 10. Here's how you can solve it:
Step 1: Identify the formula for the sum of an A.P.
The formula for the sum (S) of n terms in an A.P. with first term (a) and common difference (d) is:
S = n/2 * (a + l)
where l is the last term of the A.P.
Step 2: Determine the total number of terms (n) and middle term position (m)
We are given that n = 13 (total number of terms) and a6 = 10 (middle term value). Since 10 is the middle term, m = 6 (middle term position).
Step 3: Express l (last term) in terms of a and d
We know a6 = 10 and m = 6. Since the middle term is the average of the first and last terms, we can write:
a6 = (a + l) / 2
Substituting the known values:
10 = (a + l) / 2
Solving for l:
l = 20 - 2a
Step 4: Substitute l into the formula and solve for S
We know n = 13, a6 = 10, and l = 20 - 2a. Substituting these values into the formula for S:
S = 13/2 * (a + (20 - 2a))
Simplifying and combining like terms:
S = 13 * (10 + a)
Step 5: Find a using the middle term information (a6 = 10)
We are also given that a6 = 10. Since a6 = (n + 1)/2 * a + d, we can write for this specific case:
10 = (6 + 1)/2 * a + d
Simplifying:
10 = 3.5a + d
Solving for a:
a = 2.85
Step 6: Substitute a back into the equation for S and solve
We found a = 2.85. Plugging this value into the equation for S:
S = 13 * (10 + 2.85)
S = 169
Therefore, the sum of the 13 terms in the A.P. is 169.