An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
Sum of N Terms of A.P.
For any progression, the sum of n terms can be easily calculated. For an AP, the sum of the first n terms can be calculated if the first term and the total terms are known. The formula for the arithmetic progression sum is explained below:
Consider an AP consisting of “n” terms.
| S = n/2[2a + (n − 1) × d] |
Formula to find the sum of AP when first and last terms are given as follows:
| S = n/2 (first term + last term) |
Solution
Let’s consider a1, a2, … am be m numbers such that 1, a1, a2, … am, 31 is an A.P.
And here,
a = 1, b = 31, n = m + 2
So, 31 = 1 + (m + 2 – 1) (d)
30 = (m + 1) d
d = 30/ (m + 1) ……. (1)
Now,
a1 = a + d
a2 = a + 2d
a3 = a + 3d …
Hence, a7 = a + 7d
am–1 = a + (m – 1) d
According to the question, we have
On substituting the known values we get m= 14
