Download App

Binomial Theorem — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSBinomial Theorem1 Mark
Question
Using binomial theorem, prove that 6n–5n always leaves remainder 1 when divided by 25.
✓

Answer

For two numbers a and b if we can find numbers q and r such that a = bq + r, then we say that b divides a with q as quotient and r as remainder. Therefore, in order to show that $n^6$– 5n leaves remainder 1 when divided by 25, now we have to prove that $n^6$– 5n = 25k + 1, where k is some natural number.
Now,we have
(1 + a)n = nC0 + nC1a + nC2a2 + ... + nCnan
For a = 5, we obtain
(1 + 5)n = nC0 + nC15 + nC252 + ... + nCn5n
i.e. (6)n = 1 + 5n + 52.nC2 + 53.nC3 + ... + 5n
i.e. n 6n – 5n = 1 + 52 (nC2 + nC35 + ... + 5n-2)
or 6n – 5n = 1 + 25 (nC2 + 5.nC3 + ... + 5n-2)
or 6n – 5n = 25k+1 where k = nC2 + 5.nC3 + ... + 5n–2.
Therefore,This shows that when divided by 25, 6n – 5n leaves remainder 1.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Binomial Theorem→Binomial Theorem — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

In a $\triangle\text{ABC},$ if $\text{b}=\sqrt{3},$ c = 1 and $\angle\text{A}=30^{\circ},$ find a.
→
Translate the following statement into symbolic form:
Rahul passed in Hindi and English.
→

Find the number of terms in the expansion of $(\text{a}+\text{b}+\text{c})^{\text{n}}.$

→
If a set contains n elements, then write the number of elements in its power set.
→
If the lengths of semi-major and semi-minor axes of an ellipse are 2 and $\sqrt{3}$ and their corresponding equations are y - 5 = 0 and x + 3 = 0, then write the equation of the ellipse.
→
Write $-1 + \text{i}\sqrt{3}$ in polar form.
→
Are the following pairs of sets equal? Give reasons.
A = {x : x is a letter of the word " WOLF"};
B = {x : x is a letter of the word " FOLLOW"}.
→
If $(1+\text{i})\text{z}=(1-\text{i})\bar{\text{z}},$ then show that $\text{z}=-\text{i}\bar{\text{z}}$
→
Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A $\cap$ B
→
Write the value of $\cos1^\circ+\cos2^\circ+\cos3^\circ+\ ...\ +\cos180^\circ.$
→