Historical Sequence of the Peshwas:

I. Balaji Vishwanath
II. Balaji Baji Rao
III. Baji Rao-I
IV. Narayan Rao
V. Madhav Rao-I

Explanation:

I. Balaji Vishwanath:
- Balaji Vishwanath was appointed as the first Peshwa of the Maratha Empire in 1713 by the Maratha King Shahu.
- He served as the Peshwa for nearly 20 years until his death in 1720.
- During his tenure, he consolidated the power of the Peshwas and played a crucial role in the expansion of the Maratha Empire.

II. Balaji Baji Rao:
- Balaji Baji Rao, also known as Nana Saheb, succeeded his father as the Peshwa in 1740.
- He played a significant role in expanding the Maratha Empire and strengthening its administration.
- Balaji Baji Rao successfully led the Marathas in the Third Battle of Panipat in 1761, but the Marathas suffered a major defeat.
- He died in 1761, a few months after the battle.

III. Baji Rao-I:
- Baji Rao-I, also known as Thorale Baji Rao, was the third Peshwa of the Maratha Empire.
- He succeeded his father, Balaji Vishwanath, in 1720 and played a crucial role in expanding the Maratha territories.
- Baji Rao-I is considered one of the greatest Peshwas and is known for his military campaigns against the Mughals and other regional powers.
- He died in 1740, and his son, Balaji Baji Rao, succeeded him as the Peshwa.

IV. Narayan Rao:
- Narayan Rao was the fifth Peshwa of the Maratha Empire, succeeding his uncle, Madhav Rao-I, in 1772.
- His reign was marked by internal conflicts and power struggles within the Maratha Empire.
- Narayan Rao was assassinated in 1773, allegedly by his uncle Raghunath Rao, which resulted in a succession crisis.

V. Madhav Rao-I:
- Madhav Rao-I was the fourth Peshwa of the Maratha Empire, succeeding his father, Balaji Baji Rao, in 1761.
- He played a crucial role in stabilizing the Maratha Empire after the defeat in the Third Battle of Panipat and worked towards strengthening its administration.
- Madhav Rao-I is known for his reforms and efforts to revive the Maratha Empire.
- He died in 1772, and his nephew Narayan Rao succeeded him as the Peshwa.

Therefore, the correct historical sequence of the Peshwas is:
I. Balaji Vishwanath
III. Baji Rao-I
II. Balaji Baji Rao
IV. Narayan Rao
V. Madhav Rao-I