If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is
A. $$\frac{{ab}}{{2\left( {b - a} \right)}}$$
B. $$\frac{{ab}}{{b - a}}$$
C. $$\frac{{3ab}}{{2\left( {b - a} \right)}}$$
D. None of these
Answer: Option C
Solution (By Examveda Team)
First term (a1) = aSecond term (a2) = b
and last term (l) = 2a
∴ d = Second term - First term = b - a
∴ l = an = a + (n - 1)d
$$\eqalign{ & \Rightarrow 2a = a + \left( {n - 1} \right)\left( {b - a} \right) \cr & \Rightarrow \left( {n - 1} \right)\left( {b - a} \right) = a \cr & \Rightarrow n - 1 = \frac{a}{{b - a}} \cr & \Rightarrow n = \frac{a}{{b - a}} + 1 \cr & \Rightarrow n = \frac{{a + b - a}}{{b - a}} \cr & \Rightarrow n = \frac{b}{{b - a}} \cr & \therefore {S_n} = \frac{n}{2}\left[ {a + l} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{b}{{2\left( {b - a} \right)}}\left[ {a + 2a} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{3ab}}{{2\left( {b - a} \right)}} \cr} $$
This Question Belongs to Arithmetic Ability >> Progressions
Step-by-Step Derivation
Let's use this formula and solve for the one thing we don't know: n.
Step 1: Substitute the known values into the formula.
Last Term: 2a
First Term: a
Common Difference (d): b - a
Plugging these in, we get:
2a = a + (n - 1)(b - a)
Step 2: Isolate the part with n in it.
We want to get the (n - 1) part by itself. To do that, let's subtract a from both sides of the equation.
2a - a = (n - 1)(b - a)
a = (n - 1)(b - a)
Step 3: Continue to isolate n.
Now, the (n - 1) is being multiplied by (b - a). To undo this, we divide both sides by (b - a).
a / (b - a) = n - 1
Step 4: Get n completely by itself.
To get rid of the -1, we just add 1 to both sides.
[ a / (b - a) ] + 1 = n
Step 5: Simplify into a single fraction (optional but cleaner).
To add 1, we can rewrite it using a common denominator: 1 = (b - a) / (b - a).
n = [ a / (b - a) ] + [ (b - a) / (b - a) ]
Now, add the tops (numerators):
n = (a + b - a) / (b - a)
The a and -a on top cancel out, leaving us with the final result:
n = b / (b - a)
Next
The easiest sum formula is: Sum = (n / 2) * (First Term + Last Term)
Plug in the values we know:
Sum = (n / 2) * (a + 2a)
Sum = (n / 2) * (3a)
Now, just replace n with what we found in Step 1:
Sum = ( b / (b - a) / 2 ) * (3a)
Putting it all together gives:
Sum = 3ab / (2(b - a))
Therefore, the correct option is C.