The correct answer is option A: I, III, II, V, IV. Let's understand the historical sequence of the mentioned Peshwas:

I. बालाजी विश्वनाथ (Balaji Vishwanath)
- Balaji Vishwanath was the first Peshwa of the Maratha Empire. He served as the Peshwa from 1713 to 1720.
- Balaji Vishwanath played a crucial role in expanding the Maratha Empire and establishing the Peshwa system as the administrative structure of the empire.

III. बाजीराव प्रथम (Bajirao I)
- Bajirao I, also known as Bajirao Ballal, was the second Peshwa of the Maratha Empire. He served as the Peshwa from 1720 to 1740.
- Bajirao I is considered one of the greatest warriors and military strategists in Indian history. He expanded the Maratha Empire through numerous military campaigns and established Maratha dominance in northern India.

II. बालाजी बाजीराव (Balaji Bajirao)
- Balaji Bajirao, also known as Nana Saheb, was the third Peshwa of the Maratha Empire. He served as the Peshwa from 1740 to 1761.
- Balaji Bajirao continued the military expansion initiated by his father, Bajirao I. He played a significant role in expanding Maratha control over various parts of India, including Gujarat and Bengal.

V. माधव राव (Madhavrao)
- Madhavrao, also known as Madhavrao Peshwa, was the fourth Peshwa of the Maratha Empire. He served as the Peshwa from 1761 to 1772.
- Madhavrao's reign was marked by a period of stability and consolidation of the Maratha Empire. He focused on administrative reforms and strengthening the empire's infrastructure.

IV. नारायण राव (Narayan Rao)
- Narayan Rao was the fifth Peshwa of the Maratha Empire. He served as the Peshwa from 1772 to 1773.
- Narayan Rao's reign was short-lived and tumultuous. He faced internal conflicts and was assassinated, leading to a power struggle within the Maratha Empire.

In summary, the historical sequence of the mentioned Peshwas is: I, III, II, V, IV.