Multiple choice

Suppose a series of $n$ terms is given by $\displaystyle S_{n}=a_{1}+a_{2}+...+a_{n}$ Then $\displaystyle S_{n-1}=a_{1}+a_{2}+...+a_{n-1}\left ( \forall \ n > 1 \right )$ and then we can write $\displaystyle t_{n}=S_{n}-S_{n-1} \ \forall n\geq 2$ for $ n=1 $ we have $\displaystyle S_{1}=t_{1}$ On the basis of above information answer the following questions If sum to $ n$ terms of a series is $\displaystyle an^{2}+bn$ where $a$, $b$ are constants then $ 5^{th} $ term of the series is

  1. $\displaystyle 5a+b$
  2. $\displaystyle 9a+b$
  3. $\displaystyle 9a+2b$
  4. $\displaystyle 7a+b$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sn = an^2 + bn. t(n) = Sn - S(n-1). t(5) = S(5) - S(4). S(5) = a(25) + 5b. S(4) = a(16) + 4b. t(5) = (25a + 5b) - (16a + 4b) = 9a + b.

AI explanation

Using the formula for the nth term of a series when the sum to n terms is given, tn = Sn - S(n-1), we substitute the given sum function an^2 + bn. This yields (an^2 + bn) - (a(n-1)^2 + b(n-1)), which simplifies to a(2n - 1) + b. Plugging in n = 5, we get 9a + b.