Question

the first term of an AP is 5 and the last term is 45 and the sum is 400.find the number of term and common difference. please give step by step calculation on how to find the difference,

Answer 1

$\large\boxed{\fcolorbox{blue}{yellow} {Answer}}$

The number of terms is 160/11.

The common difference is 2.42

$\large\boxed{\fcolorbox{blue}{yellow} {Step - by - step explanation }}$

Let the first term of the A. P. be 'a' and common difference be 'd'.

Here,

a = 5

tn = 45

Sn = 400

We have to find number of terms (n) and common difference (d).

We know that,

$sn \: = \: \frac{n}{2} (a + l) \\ \\ ∴ \: 400 = \frac{n}{2} (2 \times 5 + 45) \\ \\ ∴ \: 400 = \frac{n}{2} (10 + 45) \\ \\ ∴ \: 400 = \frac{n}{2} \times 55 \\ \\ ∴ \: 400 \times 2 = 55n \\ \\ ∴ \: n = \frac{400 \times 2}{55} \\ \\ ∴ \: n = \frac{80 \times 2}{11} \\ \\ ∴ \: n = \frac{160}{11}$

Now,

$tn = a + (n - 1)d \\ \\ ∴ \: 45 = 5 + ( \frac{160}{11} - 1) \times d \\ \\ ∴ \: 45 = 5 + ( \frac{160 - 11}{11} ) \times d \\ \\ ∴ \: 45 = 5 + \frac{149}{11} \times d \\ \\ ∴ \: 45 = \frac{55 + 149}{11} \times d \\ \\ ∴ \: 45 = \frac{204}{11} \times d \\ \\ ∴ \: 45 \times 11 = 204 \times d \\ \\ ∴ \: \frac{45 \times 11}{204} = d \\ \\ ∴ \: d \: = 2.42$

∴ Ans. : The number of terms (n) = 160/11 and common difference (d) = 2.42

Hope it helps!

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