Which term of the AP \(-2,-7,-12,\ldots\) is \(-77\)? Find the sum up to that term.
16th term; sum = −632
Step 1: Identify the first term and common difference.
The AP is: \(-2, -7, -12, \ldots\)
So, first term \(a = -2\).
Common difference \(d = -7 - (-2) = -7 + 2 = -5\).
Step 2: Use the nth term formula of an AP.
The nth term is given by: \(a_n = a + (n-1)d\).
We are told that \(a_n = -77\).
So, \(-77 = -2 + (n-1)(-5)\).
Step 3: Solve for n.
\(-77 = -2 - 5(n-1)\)
\(-77 + 2 = -5(n-1)\)
\(-75 = -5(n-1)\)
Divide both sides by -5:
\(n-1 = 15\)
So, \(n = 16\).
Therefore, \(-77\) is the 16th term of the AP.
Step 4: Find the sum up to 16 terms.
Sum of first n terms is given by: \(S_n = \dfrac{n}{2}[a + a_n]\).
Here, \(n = 16, a = -2, a_{16} = -77\).
So, \(S_{16} = \dfrac{16}{2}[(-2) + (-77)]\).
\(S_{16} = 8[-79]\)
\(S_{16} = -632\).
Final Answer: The required term is the 16th term and the sum up to it is \(-632\).