Question

The first term of an A.P is 5. the
2 points
last term is 45, & the sum is 400. The
number of terms & the common
difference respectively is:
a) 4 & 2/3
b) 2/3 & 4
c) 16 & 8/3
d) 8/3 & 16​

Answer 1

Answer

• First term of an AP is 5
• Last term = 45
• Sum of all the terms = 400
• Number of terms = ?
• Common Difference = ?

━━━━━━━━━━━━━━

• First we shall find the number 9f terms in the AP using the formula n/2 {a+l}. So then we shall substitute this in the formula to find the nth term which will give us the value of the common difference (d)

$\sf :\implies s_n = \dfrac{n}{2} \bigg\lgroup First Term + Last Term \bigg\rgroup$

$\sf :\implies 400 = \dfrac{n}{2} \bigg\lgroup 5+45 \bigg\rgroup$

$\sf :\implies 400 = \dfrac{n}{2} \times 50$

$\sf :\implies 400 = n\times 25$

$\sf :\implies \dfrac{400}{25} = n$

$:\implies \underline{\boxed{\red{\mathfrak{n = 16}}}}$

• Next substitute the given values in the formula to find the nth term to find the the value of common difference as we now have the value of a, n & aⁿ

$\sf :\implies a_n = a+(n-1)d$

$\sf :\implies a_16 = a+(16-1)d$

$\sf :\implies 45 = 5+(15)d$

$\sf :\implies 45-5 = 15d$

$\sf :\implies 40 = 15d$

$\sf :\implies \dfrac{40}{15} = d$

$:\implies \underline{\boxed{\pink{\mathfrak{n = \dfrac{8}{3}}}}}$

$\displaystyle\therefore\:\underline{\textsf{The Answer is \textbf{Option C}}}$

Answer 2

Answer:

$\underline{ \sf{ \underline{☃Given:}}}$

• First term (a) = 5
• Last term (l) = 45
• Sum of terms (Sn) = 400

$\underline{ \sf{ \underline{☃Find:}}}$

• Number of terms?
• Common difference?

$\underline{ \sf{ \underline{☃Solution:}}}$

☘ Number of terms:-

We know that ⤵

${ \boxed{ \sf{ s_{n} = \frac{n}{2}(a + l) }}}$

${ \to{ \sf{400 = \frac{n}{2}(5 + 45) }}}$

${ \to{ \sf{400 = \frac{n}{2}(50) }}}$

${ \to{ \sf{400 = 25n}}}$

${ \to{ \sf{n = \frac{400}{25} }}}$

${ \to{ \sf{n = 16}}}$

So, Number of terms = 16

☘ Common Difference:-

By using the formula ⤵

${ \boxed{ \sf{ s_{n} = \frac{n}{2}(2a + n - 1) d}}}$

${ \to{ \sf{400 = \frac{16}{2} (2 \times 5 + (16 - 1)d)}}}$

${ \to{ \sf{400 = 8(10 + 15d)}}}$

${ \to{ \sf{ \frac{400}{8} = 10 + 15d}}}$

${ \to{ \sf{50 = 10 + 15d}}}$

${ \to{ \sf{50 - 10 = 15d}}}$

${ \to{ \sf{15d = 40}}}$

${ \to{ \sf{d = \frac{40}{15} = \frac{8}{3} }}}$

So, common difference = 8/3

Therefore,

Number of terms = 16

Common difference = 8/3

Step-by-step explanation:

"Option C is your answer"