The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Step-by-Step Solution
Key Concept: Use the sum formula for an arithmetic progression $S = \frac{n}{2}(a + l)$ to find the number of terms $n$, and then use the relation $l = a + (n-1)d$ to determine the common difference $d$.
1. Given data\
First term $a = 5$, last term $l = 45$, sum $S = 400$.\
2. Find the number of terms $n$ using the sum formula for an AP:\
$$S = \frac{n}{2}(a + l)$$\
Substituting the known values:\
$$400 = \frac{n}{2}(5 + 45)$$\
$$400 = \frac{n}{2}\times 50$$\
$$400 = 25n$$\
$$n = \frac{400}{25} = 16.$$\
3. Find the common difference $d$ using the relation between the first term, last term and number of terms:\
$$l = a + (n-1)d$$\
$$45 = 5 + (16-1)d$$\
$$45 = 5 + 15d$$\
$$15d = 45 - 5 = 40$$\
$$d = \frac{40}{15} = \frac{8}{3}.$$\
4. Result\
Number of terms $n = 16$\
Common difference $d = \dfrac{8}{3}$ (approximately $2.67$).
Correct Answer: Number of terms $n = 16$, common difference $d = \dfrac{8}{3}$.