Question

What is the sum of first 7 terms ?

Answer 1

We are given a formula for the sum of the first n terms of an arithmetic sequence, Sₙ = n² + 6n. We need to find the sum of the first 7 terms.

To find the sum of the first 7 terms, S₇, we just need to substitute n=7 into the given formula.

The given formula is Sₙ = n² + 6n.
Substitute n=7:
S₇ = (7)² + 6(7) S₇ = 49 + 42 S₇ = 91 The sum of the first 7 terms is 91.

Answer 2

We are given the sum of the terms, Sₙ = 315, and we need to find the number of terms, n.

We set the given formula for the sum equal to 315 and solve the resulting quadratic equation for n.
n² + 6n = 315 First, write the equation in standard quadratic form (an² + bn + c = 0):
n² + 6n - 315 = 0 We can solve this by factoring. We need two numbers that multiply to -315 and have a sum of +6.
Let's find the factors of 315: 315 = 5 × 63 = 5 × 9 × 7 = 3 × 3 × 5 × 7.
We can group the factors to find a pair with a difference of 6. Let's try (3 × 5) = 15 and (3 × 7) = 21.
The difference between 21 and 15 is 6. To get a sum of +6, we need +21 and -15.
So, the equation can be factored as:
(n + 21)(n - 15) = 0 This gives two possible solutions for n:
n + 21 = 0 n = -21
n - 15 = 0 n = 15
Since n represents the number of terms in a sequence, it must be a positive integer. Therefore, we discard n = -21.

15 terms must be added to get a sum of 315.