If (20)^19 + 2 (21) (20)^18 + 3 (21)^2 (20)^17 + + 20 (21)^19 = k (20)^19, then k is equal to:

Given: S = (20)^19 + 2 (21) (20)^18 + 3 (21)^2 (20)^17 + + 20 (21)^19 = k (20)^19 Find: k The provided solution states that on calculating the given series, we obtain: k = 400 Therefore, the value of k is 400. Given: S = (20)^19 + 2 (21) (20)^18 + + 20 (21)^19 Find: k such that S = k (20)^19 From the solution: - We use the given series and simplifications. - By calculating this series, we find that k = 400. Thus, the required numerical value is 400.

If (20)^19 + 2 (21) (20)^18 + 3 (21)^2 (20)^17 + + 20 (21)^19 = k… — Solution | JEE Mathematics

NVAMediumJEE 2023Sum of Series

If $\left(20\right)^{19} + 2\left(21\right)\left(20\right)^{18} + 3\left(21\right)^2 \left(20\right)^{17} + \cdots + 20\left(21\right)^{19} = k\left(20\right)^{19}$ , then $k$ is equal to _____:

Answer

Correct answer:400

Step-by-step solution

Standard Method

Given:

S = \left(20\right)^{19} + 2\left(21\right)\left(20\right)^{18} + 3\left(21\right)^2 \left(20\right)^{17} + \cdots + 20\left(21\right)^{19} = k\left(20\right)^{19}

Find: $k$

The provided solution states that on calculating the given series, we obtain:

k = 400

Therefore, the value of $k$ is $400$ .

Explanation from Extracted

Given:

S = \left(20\right)^{19} + 2\left(21\right)\left(20\right)^{18} + \cdots + 20\left(21\right)^{19}

Find: $k$ such that $S = k\left(20\right)^{19}$

From the solution:

We use the given series and simplifications.
By calculating this series, we find that $k = 400$ .

Thus, the required numerical value is $400$ .

Common mistakes

A common mistake is to treat the expression as a simple geometric progression. This is wrong because the coefficients $1, 2, 3, \ldots, 20$ also vary. Instead, recognize it as a weighted series.
Another mistake is to miss the factor $\left(20\right)^{19}$ on the right-hand side. This leads to comparing the series directly with $k$ instead of with $k\left(20\right)^{19}$ . Always identify the common power structure carefully.

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Answer

Step-by-step solution

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