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                                            Arithmetic Progression Example.

Q:If  the sum of the first 7 terms of an AP is 49 and that of the first 17 terms is 289,find the sum of its first n terms.

                                                                  Solution:

 We know that the Sum of the first n terms of an AP is given by:

                                   Sn = (n / 2) * ((2 * a + (n-1) * d)).

Therefore, as per the first requirement,

                                          49 = (7 / 2) * ((2 * a + (7-1) * d)).

• 98 = 14a + 42d
• 7 = a + 3 d                                    (1)

Similarly, as per the second requirement,

                                    289 = (17 / 2) * ((2 * a + (17-1) * d)).

• 578 = 34a + 272d
• 17 = a + 8d (2)

Now, from (1), we have , a = 7 – 3d. Substituting this value of a in (2), we have

                          17 = 7-3d + 8d

                         Whence, 10 = 5d,

                           Thus, d = 2.

And, then a is 7 – (3 * 2) = 7 – 6 = 1

Thus, we have, a = 1 and d = 2. Now, the sum of the first n terms is

                  Sn = (n / 2) * ((2 * a + (n-1) * d)).

• Sn = (n / 2) * ((2 * 1 + (n-1) * 2)). ---- Substituting the values of a and d obtained earlier
• Sn = (n / 2) * ((2 + 2n -2))
• Sn = (n / 2) * (2n)
• Sn = n^2

Thus,

                    The sum of its first n terms = n^2.

The given sequence is an arithmetic progression (AP) with a common difference. To find this common difference, let's subtract each term from the next one:

To find which term of the arithmetic progression

To find out how many terms of the arithmetic progression 45,39,33,…45,39,33,… must be taken so that their sum is 180180, we can use the formula for the sum of the first nn terms of an arithmetic progression:

In an arithmetic progression

Let's denote the four consecutive numbers in the arithmetic progression

The difference between any two terms in an arithmetic progression (AP) is constant and equal to the common difference (d).

In this arithmetic progression, the common difference is

Let's denote the sum of the first nn terms of the first arithmetic progression