We are given that the sum of the first 7 terms is 49. Substituting n = 7 into the sum formula gives (7)/(2)(2a + 6d) = 49. Dividing both sides by 7 and multiplying by 2 yields 2a + 6d = 14, which simplifies to a + 3d = 7.

We are given that the sum of the first 7 terms is 49. Substituting n = 7 into the sum formula gives (7)/(2)(2a + 6d) = 49. Dividing both sides by 7 and multiplying by 2 yields 2a + 6d = 14, which simplifies to a + 3d = 7.

We are also given that the sum of the first 17 terms is 289. Substituting n = 17 into the formula gives (17)/(2)(2a + 16d) = 289. Since 289 = 17², dividing by 17 and multiplying by 2 gives 2a + 16d = 34, which simplifies to a + 8d = 17.

We are also given that the sum of the first 17 terms is 289. Substituting n = 17 into the formula gives (17)/(2)(2a + 16d) = 289. Since 289 = 17², dividing by 17 and multiplying by 2 gives 2a + 16d = 34, which simplifies to a + 8d = 17.

Now we find the sum of the first 25 terms by setting n = 25, a = 1, and d = 2 in the formula S_n = (n)/(2)[2a + (n-1)d]. We get S_25 = (25)/(2)[2 + 48] = (25)/(2) × 50 = 25 × 25 = 625. Note that for this AP, S_n = n².

Now we find the sum of the first 25 terms by setting n = 25, a = 1, and d = 2 in the formula S_n = (n)/(2)[2a + (n-1)d]. We get S_25 = (25)/(2)[2 + 48] = (25)/(2) × 50 = 25 × 25 = 625. Note that for this AP, S_n = n².

the sum of the first 7 terms of an ap is 49 and some of the first 17 times is 289 find the sum of the first 25 terms

Answer: 625

Step-by-step solution

Step 1: State the formula for the sum of first n terms

Let $a$ be the first term and $d$ be the common difference of the given AP. The sum of the first $n$ terms of an arithmetic progression is given by the formula $S_n = \frac{n}{2}[2a + (n - 1)d]$ .

Step 2: Form equation using the sum of the first 7 terms

We are given that the sum of the first $7$ terms is $49$ . Substituting $n = 7$ into the sum formula gives $\frac{7}{2}(2a + 6d) = 49$ . Dividing both sides by $7$ and multiplying by $2$ yields $2a + 6d = 14$ , which simplifies to $a + 3d = 7$ .

Step 3: Form equation using the sum of the first 17 terms

We are also given that the sum of the first $17$ terms is $289$ . Substituting $n = 17$ into the formula gives $\frac{17}{2}(2a + 16d) = 289$ . Since $289 = 17^2$ , dividing by $17$ and multiplying by $2$ gives $2a + 16d = 34$ , which simplifies to $a + 8d = 17$ .

Step 4: Solve the system of linear equations for a and d

Subtracting equation (1), $a + 3d = 7$ , from equation (2), $a + 8d = 17$ , eliminates $a$ and leaves $5d = 10$ , so $d = 2$ . Substituting $d = 2$ back into $a + 3d = 7$ gives $a + 6 = 7$ , which gives $a = 1$ .

Step 5: Calculate the sum of the first 25 terms

Now we find the sum of the first $25$ terms by setting $n = 25$ , $a = 1$ , and $d = 2$ in the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ . We get $S_{25} = \frac{25}{2}[2 + 48] = \frac{25}{2} \times 50 = 25 \times 25 = 625$ . Note that for this AP, $S_n = n^2$ .

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