# If a, b, c an d are in G.P. Show that: (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2}) = (ab + bc + cd)^{2}

**Solution:**

If a, b, c and d are in G.P.

Therefore,

bc = ad ....(1)

b^{2} = ac ....(2)

c^{2} = bd ....(3)

We need to prove that, (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2}) = (ab + bc + cd)^{2}

Since,

RHS = (ab + bc + cd)^{2}

= (ab + ad + cd)^{2} [Using (1)]

= [ab + d (a + c)]^{2}

= a^{2}b^{2} + 2abd (a + c) + d ^{2} (a + c)2

= a^{2}b^{2} + 2a^{2}bd + 2acbd + d ^{2} (a^{2} + 2ac + c^{2} )

= a^{2}b^{2} + 2a^{2}c^{2} + 2b^{2}c^{2} + d ^{2}a^{2} + 2d ^{2}b^{2} + d ^{2}c^{2} [Using (1) and (2)]

= a^{2}b^{2} + a^{2}c^{2} + a^{2}c^{2} + b^{2}c^{2} + b^{2}c^{2} + d ^{2}a^{2} + d ^{2}b^{2} + d ^{2}b^{2} + d ^{2}c^{2}

= a^{2}b^{2} + a^{2}c^{2} + a^{2}d ^{2} + b^{2} x b^{2} + b^{2}c^{2} + b^{2}d ^{2} + c^{2}b^{2} + c^{2} x c^{2} + c^{2}d ^{2}

Using (2) and (3) and rearranging the terms

RHS = a^{2} (b^{2} + c^{2} + d ^{2}) + b^{2} (b^{2} + c^{2} + d ^{2}) + c^{2} (b^{2} + c^{2} + d ^{2})

= (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2})

= LHS

Thus, (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2}) = (ab + bc + cd)^{2} Proved

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.3 Question 25

## If a, b, c an d are in G.P. Show that: (a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2}) = (ab + bc + cd)^{2}

**Summary:**

Given that a, b, c, d in G.P we showed that a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d ^{2}) = (ab + bc + cd)^{2}

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