How many terms are there in an AP whose first term and 6th term are -12 and 8 respectively, and the sum of all its terms is 120?

There are 12 terms in the Arithmetic Progression (AP).

Here's a breakdown of how we arrive at that answer:

Let's define the following:

• 'a' as the first term of the AP.
• 'd' as the common difference between the terms.
• 'n' as the number of terms.
• 'Sₙ' as the sum of the first n terms.

We are given:

• a = -12 (first term)
• The 6th term is 8. In AP, the nth term is given by: aₙ = a + (n-1)d
• Therefore, the 6th term, a₆ = a + 5d = 8
• We know a = -12, so: -12 + 5d = 8
• Solving for 'd', we get: 5d = 20, thus d = 4
• The sum of n terms of an AP is given by: Sₙ = (n/2) [2a + (n-1)d]
• We are given that Sₙ = 120

Now, let’s substitute the known values into the sum formula:

120 = (n/2)[2(-12) + (n-1)4]
120 = (n/2)[-24 + 4n - 4]
120 = (n/2)[4n - 28]
240 = n(4n - 28)
240 = 4n² - 28n
0 = 4n² - 28n - 240
0 = n² - 7n - 60

We solve this quadratic equation:
0 = (n - 12)(n + 5)

The possible values of n are 12 or -5.
Since the number of terms cannot be negative, n = 12

Therefore, there are 12 terms in the AP.

• First Term (a): -12
• Sixth Term (a₆): 8
• Common Difference (d): 4
• Sum of terms (Sₙ): 120
• Number of terms (n): 12

Reference:
The answer is confirmed to be 12 terms, as stated in the provided reference: "Answer: there are 12 terms in the AP. 27-Jan-2022"